Consider the Cohen forcing, and suppose that $\dot x,\dot y$ are names for reals, which are not in the ground model (i.e. $1$ forces that neither is in the ground model).

Can we always find an automorphism mapping $\dot x$ to $\dot y$?

Of course, this depends on how you consider the Cohen forcing, so let's take the "richest" way, and consider it as the countable atomless Boolean algebra.