Groups containing locally maximal product-free sets of size $4$

Every locally maximal product-free set $S$ in a finite group $G$ satisfies $G=S\cup SS \cup S^{-1}S \cup SS^{-1}\cup \sqrt{S}$, where $SS=\{xy\mid x,y\in S\}$, $S^{-1}S=\{x^{-1}y\mid x,y\in S\}$, $SS^{-1}=\{xy^{-1}\mid x,y\in S\}$ and $\sqrt{S}=\{x\in G\mid x^2\in S\}$. To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set $S$ in a finite abelian group satisfy $|\sqrt{S}|\leq 2|S|$. This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size $4$, continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size $4$, and conclude with a conjecture on the size $4$ problem as well as an open problem on the general case.