Hash length extension attack - SHA256 to 512 - impossible, correct? Just want to be sure about something before I sign off on a method we're about to put into place.

We have a secret, to which a user-defined key is appended. The user can see the SHA256 hash of the secret.

The secret and the user-key is then hashed with SHA512 to obtain results.

Given the ending hash is SHA512, and the user only knows the SHA256, a hash length extension attack would be impossible, correct? As there is no way to convert a 256 to 512, or to backtrack the 256?

From every angle I can see it, it shouldn't be possible. But want to confirm with those more well versed than I. Thanks.
No, there is no known way.

It would actually be rather surprising if there were even a theoretical way; the SHA-256 and the SHA-512 compression functions are rather different (for one, one works with 32 bit words and the other works with 64 bit words); one wouldn't expect them to share any sort of relation.