Let $G$ be a $\{2,3\}$-group and $\lvert G\rvert=2^\alpha\cdot3^\beta$. For $p\in\{2,3\}$, define $$ \nu_p(G)\mathrel{:=}\min\left\{\log_p\left(\frac{\lvert G\rvert}{\chi(1)}\right)_p \mathrel{\bigg\vert} \chi\in\operatorname{Irr}(G)\right\}, $$ where $\operatorname{Irr}(G)$ is the set of all irreducible $\mathbb{C}$-characters of $G$.

Suppose that $\nu_2(G)=1$, $\nu_3(G)=0$. We want to study this group $G$. Especially, we want to know the answer to:

QUESTION: Are there two numbers $M$ and $N$ such that $\alpha<M$ and $\beta<N$ for all such groups $G$?