Let $ G $ be a (non-abelian) finite simple group. An extension $ G\cdot m $ is nontrivial if it is not isomorphic to the direct product $ G \times m $. Suppose that there exists a nontrivial extension $ G\cdot m $ where $ m $ is a cyclic group of order $ m $. Prove that for every $ n $ divisible by $ m $ there exists a nontrivial extension $ G.n $.
Are the assumptions on $ G $ necessary? Or is it even true more generally that if there exists a nontrivial extension $ G\cdot m $ where $ m $ is a cyclic group of order $ m $ then for every $ n $ divisible by $ m $ there exists a nontrivial extension $ G.n $.