In a previous paper of the first author a procedure was developed for
counting the components of a graph through the knowledge of the components of
its quotient graphs. We apply here that procedure to the proper power graph
P0​(G) of a finite group G, finding a formula for the number
c(P0​(G)) of its components which is particularly illuminative when
G≤Sn​ is a fusion controlled permutation group. We make use of the proper
quotient power graph P0​(G), the proper order graph
O0​(G) and the proper type graph T0​(G). We show that
all those graphs are quotient of P0​(G) and demonstrate a strong
link between them dealing with G=Sn​. We find simultaneously
c(P0​(Sn​)) as well as the number of components of
P0​(Sn​), O0​(Sn​) and
T0​(Sn​)