I found the solution, and I am posting here in case it could be useful for others in the future, too:

Basically, I didn't manage to import properly earlier because  "Segwit only allows compressed public keys, but Armory only uses uncompressed public keys. In order to use Segwit, Armory will compress the keys on-the-fly, but the export will still give you the uncompressed version."

So I simply needed to get the compressed version of the private key (namely, the compressed WIF private key) from the (uncompressed) WIF private key that Armory was giving me.
I used a local version of bitaddress.org (taken from the repo at  https://github.com/pointbiz/bitaddress.org  )  to do that.

Credits and thanks to achow101 for telling me that (and full detailed instructions in case other people might need it):
https://bitcointalk.org/index.php?topic=2689834.msg27503623#msg27503623


Edit:  Typo.

Yes, it did work!   

Much gratitude for your help and for sharing your knowledge, achow101.  You made my day!

And I feel I understand a tiny bit more of the workings of Bitcoin now.  Still a long way to go...

Perfect!  It looks exactly like what I was needing!

I should be able to convert the private keys from the uncompressed to the compressed format using a local version of bitaddress.org - now that I know this is what I need to do!

Will do that and report here.
Many thanks for your much appreciated help.

If that may help, this is for a P2SH-P2WPKH private key exported from Armory v0.96.3.99 (0.96.4 RC1).

I had tried to do that on the console ("addwitnessaddress [the_legacy_address_that_appeared]", but it gave me the error:

"Public key or redeemscript not known to wallet, or the key is uncompressed (code -4)"

(I did not rescan, as it seemed the addwitnessaddress command wasn't successful.)

There's something I do not know or understand about segwit's functionning, and so I have no clue how to progress forward with that.
What would that mean?

I agree with you, however the issue is that I am already using a segwit (P2SH-P2WPKH) address from another wallet, and I want to import that into Core.

I'm sure there must be a way to do that with Core 0.15.1, I just have to find out how!

Some more specific information, gathered from another thread where goatpig wrote:

Source:  https://bitcointalk.org/index.php?topic=2679892.msg27479932#msg27479932

It seems that what I need to do is to have the Bitcoin core client to "instantiate the associated address as nested SW".

How could I do that?

Thank you for that, I understand more now, and I know in which direction to continue searching.

How to transfer/import a segwit (P2SH-P2WPKH) address into Bitcoin Core, so that I can send the coins from Bitcoin Core (the original client)?

I have created Segwit addresses (P2SH-P2WPKH) on another client (Armory) and received funds there, and now I want to "import" those Segwit addresses with the Bitcoin core client, so that I can spend the funds from the Bitcoin core client.  I couldn't find how to do that.

With legacy (P2PKH) addresses ("1-" addresses), I can simply import the private keys into Bitcoin core, and it all works fine.  But if I try to import the private key of a segwit (P2SH-P2WPKH) address into Core (through the console) it gives me a legacy "1" address that doesn't correspond to my segwit address - and which doesn't contain my coins, obviously.

What am I doing wrong here, and how could I do that?

Any help would be much appreciated.

Thank you for your answer - much appreciated.

(Just to clarify, when I mention "Core", I mean the original Bitcoin client/wallet - i.e. the same one that Armory requires to run.)

I somehow thought that being able to export private keys in a standard way was a regular feature of Armory, so that we could use them elsewhere in case of need.

My understanding of Segwit is far from being complete, and so I had the impression that I needed to extract or export something else from Armory (the redeem script?) in order to input it in the Bitcoin core client - but I cannot find where/how to export that.


Edit:  Typo.

That's a very important piece of information I didn't know.  Thank you for that.  Cycling wallets (and not only keys), and obviously making proper backups, is an important step to do beforehand then.

I prefer to avoid using Electrum, at least for now, as I do not like the idea of being dependent and a third-party server to access coins.  I think that running full nodes is healthier for the entire network.  But thank you for suggesting that.

Back to my initial question:
How to import a segwit P2SH-P2WPKH keys into bitcoin core, as it is right now?

I am sure that goatpig knows if/how this can be done, but I do not want to send him a direct PM to ask.  It is best if those discussions are public so that it can help other people in the future too. 

No, Armory works fine.  I actually believe it is the best available wallet at the moment.

I want to do this for 2 main reasons:
1- To be sure to be able to successfully export my keys/addresses and access coins in case I run into an issue with Armory in the future.  That is, to not be 100% dependent (or "stuck") on a single system, or a single point of failure.
2- To be able to export specific keys/addresses into other bitcoin forks' clients (after having emptied those keys, obviously).

Yes, I've done that, and it simply gave me another segwit (3-) address, which was not my original segwit address.  The way I understand it (and I might be wrong) is that Core can generate a segwit address from a legacy (1-, P2PKH) address, and that's what "addwitnessaddress" does.

So it seems to me that I have to find the "original" legacy (P2PKH) address that Armory used to generate a segwit address?  But I couldn't find it anywhere within Armory.

I am pretty sure that it is possible to import segwit (P2SH-P2WPKH) addresses into Bitcoin Core, based on the conversation between goatpig and achow101 here:  https://bitcointalk.org/index.php?topic=2605402.0

How to transfer/import a segwit (P2SH-P2WPKH) address (and private key) from Armory to Bitcoin Core?

How can I transfer & import the private key of a segwit (P2SH-P2WPKH) address from Armory into Bitcoin core?

I searched quite a fair bit and couldn't manage to do it.

I can export the private key (Base58) from Armory just fine, but when I import that into Bitcoin core (through the console) it gives me a legacy "1" address that doesn't correspond to my segwit address.

It seems I need to use the command "addwitnessaddress", but I do not know which legacy "1" address I should specify, nor how to find it.


What am I doing wrong here, and how could I do that?