= Solution
A <flasque sheaf> has surjective restriction maps $\mathcal F(V)\to\mathcal F(U)$ for every pair of opens $U\subseteq V$. To prove the claim for an <injective sheaf of modules>, let $j_U:U\hookrightarrow X$ and $j_V:V\hookrightarrow X$. The natural map
$$
(j_U)_!\mathcal O_U\longrightarrow(j_V)_!\mathcal O_V
$$
is a monomorphism: its <stalks> are either the identity on $\mathcal O_{X,x}$, the map from zero to that stalk, or the zero-to-zero map. Here $j_!$ is <extension by zero> for module sheaves.
The extension-by-zero adjunction identifies
$$
\operatorname{Hom}_{\mathcal O_X}((j_U)_!\mathcal O_U,I)\cong\Gamma(U,I).
$$
The <injective object> property extends every morphism from $(j_U)_!\mathcal O_U$ to one from $(j_V)_!\mathcal O_V$. Under the displayed identification this is exactly surjectivity of $I(V)\to I(U)$. \b[Injective module sheaves are therefore flasque.] This argument works on an arbitrary <ringed space>, without Noetherian or separation assumptions.
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