A lower bound for the number of odd-degree representations of a finite group

Let $G$ be a finite group and $P$ a Sylow $2$-subgroup of $G$. We obtain both asymptotic and explicit bounds for the number of odd-degree irreducible complex representations of $G$ in terms of the size of the abelianization of $P$. To do so, we, on one hand, make use of the recent proof of the McKay conjecture for the prime 2 by Malle and Sp"{a}th, and, on the other hand, prove lower bounds for the class number of the semidirect product of an odd-order group acting on an abelian $2$-group.