For finite groups $L \leq K$ we define $d(L,K)$ to be the least $n \in \mathbb{N}$ for which there exist $a_1, \dots, a_n \in K$ such that $\langle L, a_1 \dots, a_n \rangle = K$.

Is there some $m \in \mathbb{N}$ for which the following claim holds:

Let $G$ be a finite group and let $H \leq G$ be a subgroup with $d(H,G) = 1$. Suppose that $G_0$ is a subgroup of index $2$ in $G$ such that $H \leq G_0 \leq G$. Then $d(H,G_0) \leq m$.