On the pronormality of subgroups of odd index in some extensions of finite groups

We study finite groups with the following property $(*)$: All subgroups of odd index are pronormal. Suppose that $G$ has a normal subgroup $A$ with property $(*)$, and the Sylow $2$-subgroups of $G/A$ are self-normalizing. We prove that $G$ has property $(*)$ if and only if so does $N_G(T)/T$, where $T$ is a Sylow $2$-subgroup of $A$. This leads to a few results that can be used for the classification of finite simple groups with property $(*)$.