Solution
= Solution
Under (ii), tensor the direct-product decomposition with the residue field $k$. Since $W$ is free over $\mathcal O$,
$$
k\otimes_{\mathcal O}\operatorname{End}_{\mathcal O}(W)
\cong\operatorname{End}_k(\overline W)\cong M_n(k).
$$
Hence
$$
kG\cong\operatorname{End}_k(\overline W)\times(k\otimes_{\mathcal O}B),
$$
and the reduced action map is the projection onto the first factor. This proves
$$
\boxed{\text{(ii)}\Longrightarrow\text{(iv)}.}
$$