Let me see if I can clear up your confusion. It is true that any algebraic extension of a field $F$ embeds in a given algebraic closure, but this embedding is not unique; you can compose any given embedding with an element of the Galois group of the normal closure, and if the extension isn't itself Galois then you'll even end up with embeddings that have different images. Also, it's usually considered bad form to say "the" algebraic closure of a field because algebraic closures are only unique up to isomorphism and the choice of an algebraic closure cannot be made canonically (to my knowledge). Anyway, here's what I think you wanted to say:

"If I have an extension $E/F$ and an isomorphism $\sigma : F \to F'$, I should be able to find a corresponding extension $E'/F'$ and an isomorphism $\tau : E \to E'$ which restricts to $\sigma$."

This statement is true, but again, $\tau$ is not unique. (And I wouldn't call this an "isomorphism theorem." Those refer to a specific set of theorems.) I also don't know why you give a statement about isomorphisms and then ask for a statement about homomorphisms (in the group setting). What kind of statement, exactly, are you looking for?