  Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface
of degree~$2$ and $G$ be a group acting on $X$. In this paper we study
$\Bbbk$-rationality questions for the quotient surface $X / G$. If there are no
smooth $\Bbbk$-points on $X / G$ then $X / G$ is obviously
non-$\Bbbk$-rational.
  Assume that the set of smooth $\Bbbk$-points on the quotient is not empty. We
find a list of groups, such that the quotient surface can be
non-$\Bbbk$-rational. For these groups we construct examples of both
$\Bbbk$-rational and non-$\Bbbk$-rational quotients of both $\Bbbk$-rational
and non-$\Bbbk$-rational del Pezzo surfaces of degree $2$ such that the
$G$-invariant Picard number of $X$ is $1$. For all other groups we show that
the quotient $X / G$ is always $\Bbbk$-rational.
