Finite groups whose maximal subgroups have almost odd index

A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups whose irreducible character degrees, conjugacy class sizes or indices of maximal subgroups are odd. These results have been extended to include those finite groups whose character degrees or conjugacy class sizes are not divisible by $4$. In this paper, we determine the structure of finite groups whose maximal subgroups have index not divisible by $4$. As a consequence, we obtain some new $2$-nilpotency criteria.