A week ago, I came up with an injective, but not surjective homomorphism which mapped a field to the same field: $\phi: \Q(e) \to \Q(e)$ defined by $\phi(e) = e^2$ and $\left.\phi\right_\Q = \id_{\Q}$. It isn’t surjective because $\phi[\Q(e)] = \Q(e^2) \subsetneq \Q(e)$
Obviously, this kind of mapping wasn’t defined on a finite field.
After that, I found another nonsurjective embedding which sends field $\Z_p [y]$, where $y$ is an indeterminate, to itself on Mathematics Stack Exchange.^{1}
From this, I’ve understood that why the Isomorphism Extension Theomrem doesn’t apply to transcendental field extensions.

How to prove that the Frobenius homomorphism is surjective? on Mathematics Stack Exchange. ↩