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 Späth, 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.