I am thinking the following statement is true:
Direct summand of an injective module is injective
Given $0\to A\to A'$ and $A\to C$, we compose the canonical injection to obtain $A\to C\oplus C'$ which is injective, thus extends to $A'\to C\oplus C'$. We then descend this to $A'\to C$, which is an extension of $A\to C$.
Is it correct?
Yes, the proof is correct - you only have to explain the notations ($C$ is a direct summand of an injective module, say $C \oplus C'$, etc.).
A more general statement is the following: If $\mathcal{C}$ is a category, $I \in \mathcal{C}$ is injective and $J \in \mathcal{C}$ is a retract of $I$, i.e. $\mathrm{id}_J$ factors through $I$, then $J$ is injective. In fact, if $A \to A'$ is a monomorphism and $A \to J$ is a morphism, then $A \to J \hookrightarrow I$ extends to $A' \to I$, and then $A' \to I \twoheadrightarrow J$ is an extension of $A \to J$ because $J \hookrightarrow I \twoheadrightarrow J$ is the identity.