Let $(R,\m)$ and $(S,\n)$ be commutative Noetherian local rings, and let
$\phi:R\to S$ be a flat local homomorphism such that $\m S = \n$ and the
induced map on residue fields $R/\m \to S/\n$ is an isomorphism. Given a
finitely generated $R$-module $M$, we show that $M$ has an $S$-module structure
compatible with the given $R$-module structure if and only if $\Ext^i_R(S,M)=0$
for each $i\ge 1$.
We say that an $S$-module $N$ is {\it extended} if there is a finitely
generated $R$-module $M$ such that $N\cong S\otimes_RM$. Given a short exact
sequence $0 \to N_1\to N \to N_2\to 0$ of finitely generated $S$-modules, with
two of the three modules $N_1,N,N_2$ extended, we obtain conditions forcing the
third module to be extended. We show that every finitely generated module over
the Henselization of $R$ is a direct summand of an extended module, but that
the analogous result fails for the $\m$-adic completion.