Commuting involution graphs for sporadic simple groups

Let K⩽G⩽Aut(K), where K is one of the 26 sporadic finite simple groups, and let t∈G be an involution and X=tG. The commuting involution graph C(G,X) has X as its vertex set with two distinct elements of X joined by an edge whenever they commute in G. For most of the sporadic simple groups, we compute the diameter of C(G,X) and give detailed information about the elements at a given distance from a fixed involution t.