How would the private key be the exact same, between the BTC & LTC network for the public address?

private key is just a number so that is no the place you should look. what you should look into is the type of the curve and then the script types they create. since LTC is the exact copy of bitcoin, it also uses the same curve so the same number (private key) corresponds to the same public key. and since both use the same hash functions, they both come up with the same hash. and since they also use the same script structure (OP_HASH160 <20 byte> OP_EQUAL) then the scripts are also the same. the only thing different would be address format which is the base58 encoding (first byte) which doesn't matter at all.

Looks like you are right, it can be done.

found the following.

https://www.reddit.com/r/Bitcoin/comments/byk9qc/help_needed_stupid_me_sent_litcoin_from_trezor_to/eqk63ut/Here is how to do it:

Take a Bitcoin private key in WIF (wallet import format).

Convert it to hexadecimal format.

Add "b0" in front of it and "01" at the end of it.

Calculate a double SHA256 hash of the resulting string as a hex (use

https://www.fileformat.info/tool/hash.htm and use the field Binary hash / Hex bytes).

Add first 8 symbols from the result of step 4 to the resulting string of step 3.

Convert the resulting hex string to WIF using

https://incoherency.co.uk/base58/ The result is a Litecoin private key in WIF (wallet import format).

Example:

Bitcoin private key (WIF): KzNDQwKngvExhYzGhsMhHjeacwB9L1Asgo293Kf8TPMutvfr89wv

Converting to hex format: 5de30ea8c8a415f7b8a0286a033c9ff9e94415326c405cf0ce611df1067ecec6

Add "b0" in front of it and "01" at the end of it: b05de30ea8c8a415f7b8a0286a033c9ff9e94415326c405cf0ce611df1067ecec601

First round of hash: 254609e251aeb830ce0e8c37f75b76a15795d550c410b8bdda32394c1d9a7e6d

Second round of hash: 103087d15075dd2464c0b49dabce677e36d68179fbb871cd1f846be451946e37

"b05de30ea8c8a415f7b8a0286a033c9ff9e94415326c405cf0ce611df1067ecec601" + "103087d1" = b05de30ea8c8a415f7b8a0286a033c9ff9e94415326c405cf0ce611df1067ecec601103087d1

Converting to WIF: T6CUrgcy6JDZUPd9FWJZW6BxZnpTQ6BmVzvPu8Hg2MY5QpCvzyjN

Result: we converted Bitcoin private key "KzNDQwKngvExhYzGhsMhHjeacwB9L1Asgo293Kf8TPMutvfr89wv" to Litecoin private key "T6CUrgcy6JDZUPd9FWJZW6BxZnpTQ6BmVzvPu8Hg2MY5QpCvzyjN".

Proof that it works:

Bitcoin private key: KzNDQwKngvExhYzGhsMhHjeacwB9L1Asgo293Kf8TPMutvfr89wv

Litecoin private key: T6CUrgcy6JDZUPd9FWJZW6BxZnpTQ6BmVzvPu8Hg2MY5QpCvzyjN

Use

https://www.bitaddress.org (go to "Wallet Details" and enter the Bitcoin private key).

Use

https://liteaddress.org (go to "Wallet Details" and enter the Litecoin private key).

You will see that for both Bitcoin and Litecoin the hex format private keys (and the uncompressed public keys) are identical. This means that these keys will "unlock" an identical SegWit address on both networks.