The Gill-Guillot commuting graph for sporadic and related groups

Let $G$ be a finite group and $\mathcal{C}$ a normal subset of $G$. The Gill-Guillot graph has vertices $\mathcal{C}$ and $x,y\in \mathcal{C}$ are adjacent if and only if $x$ and $y$ commute and $\{x{y}^{-1},x^{-1}y\}\cap \mathcal{C}$ is non-empty. We study the connectivity of this graph for quasisimple groups with $G\mathrm{/}Z(G)$ a sporadic simple group and for certain simple groups with exceptional Schur multiplier.