about almost maximal subgroup

Question

Let $ G $ be a finite group that there is an element $ y\in G $ such that $ G = \langle y \rangle M $for any almost maximal subgroup $ M $ of $ G $ and $ y $ is $ p $-element. ( A proper subgroup $ M $ is called an almost maximal subgroup of $ G $ if $ M $ is a maximal subgroup or $ \vert G : M \vert $ is a power of a prime number.) Which can be correct ? Why ? $ G \cong S_{4} $ , $ G \cong A_{4} $ , $ G \cong Q_{2} $ or $ G \cong B_{4} $ where $ Q_{2} \in Syl_{2}(S_{4}) $. $ B_{4} $ is Klein four-group.