Question

Let G be a finite group whose order is a unit in a ring k, and let W ⊆ V be left kG-modules. If W is a direct summand of V as k-modules, then W is a direct summand of V as kG-modules.

Answer 1

Fixing a $k$-homomorphism $f \colon V \to W$ such that $f \mid W$ is the identity, we can define $g : V \to V$ as

$$g(v) = |G|^{-1} \sum_{g \in G} \sigma_i^{-1} f(\sigma v) \text{ for } v \in V.$$

We check easily that $g$ is a $kG$-homomorphism with $g \mid W = \operatorname{Id}_W$, and so $V = W \circ \ker(g)$.