Suppose we have two automorphisms on a graph G such that each one swaps a separate pairs of vertices. Is it possible to construct (or prove the existence of) a third automorphism that swaps both pairs simultaneously?  More specifically let A and B are two automorphisms on G and let x,y,z,w are four distinct nodes in the graph, such that

A swaps x and y, i.e.  A(x) = y and A(y) = x

and

B swap w and z, i.e. B(w) = z and B(z) = w

The question that I'm trying to solve is whether there exists an automorphism C such that C swaps both x,y and w,z  i.e.

 C(x) = y and C(y) = x and C(w) = z and C(z) = w

So far my attempts failed (probably because my graph theory background is not that strong). May be the answer is straightforward so apologies if this doesn't qualify as a "research level" question. Any pointers would be highly appreciated. Thanks.