I found the quoted question in this post interesting: Confusion regarding what kind of isomorphism is intended. I don't have commenting privileges just yet, and since the question already has an accepted answer and it's purpose was not to actually receive an answer to the quoted question, I've decided to post here.
I've been able to do the first part there by using Maschke's theorem and showing that the existence of a two-sided inverse for the projection map from $V$ to a submodule forming part of a decomposition of $V$ leads to a contradiction.
The second part however, I am stumped, could someone kindly provide some guidance?