2-McKay conjecture for 2-representations

Let $H$ be a finite group, let $p$ be a fixed prime, let $Z$ be the centre of $H$, let $P$ be a Sylow $p$-subgroup, and let $N$ be its normaliser. Let $G$ be the group of automorphisms of $H$ that preserve $P$ and restrict to the identity on $Z$. For each character $\chi:Z\to{\mathbb C}^{\times}$, let

$$\Theta_H(\chi)=\Theta_{{\mathbb C}H_{p'}\otimes_{{\mathbb C}Z}{\mathbb C}(\chi)},\qquad \Theta_N(\chi)=\Theta_{{\mathbb C}N_{p'}\otimes_{{\mathbb C}Z}{\mathbb C}(\chi)}.$$

2-McKay conjecture. For each character $\chi$, the 2-representations $\Theta_H(\chi)$ and $\Theta_N(\chi)$ of $G$ are isomorphic. This is a 2-representation-theoretic strengthening of the McKay comparison between $H$ and the normaliser of a Sylow subgroup, motivated by the notion of McKay-goodness.