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 $\dot y$?


If the answer is negative, as I suspect it is, is there some reasonable condition on the names in order to have the answer yes? 

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.