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 $\dot y$?

The answer is negative, as Andreas Blass points out. But let me refine the question a lot more.

Suppose $p$ is a condition which forces that $\dot x$ and $\dot y$ are both generic with respect to the restriction of the forcing to $A$ (i.e. take only conditions whose domain is a subset of $A$ and complete that to the subalgebra of the Cohen forcing). Is there an automorphism $\pi$ such that $p\Vdash\pi\dot x=^*\dot y$?

Of course, this depends on how you consider the Cohen forcing, so let's take the "richest" way, and consider it as the completion of functions from finite sets of integers to $\{0,1\}$.