Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$

Question

Let $G$ be a group and $K$ be subgroup of order $p^a$. I am trying to show that $$|\{H \leq G : K \subset H, |H|=p^{a+b}\}| \equiv 1 \pmod{p}$$ for each b fixed & for all $a+b$ such that $p^{a+b}$ divides $|G|$.

In Sylow's Theorem, we have the option to consider conjugation by one Sylow $p$-subgroup on the set of $p$-Sylow subgroups. But here what can be done?