  Smooth and symplectic symmetries of an infinite family of distinct exotic
$K3$ surfaces are studied, and comparison with the corresponding symmetries of
the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth
finite group action is shown to be strongly restricted, and as a result,
nonsmoothability of actions induced by a holomorphic automorphism of a prime
order $\geq 7$ is proved and nonexistence of smooth actions by several $K3$
groups is established (included among which is the binary tetrahedral group
$T_{24}$ which has the smallest order). Concerning symplectic symmetries, the
fixed-point set structure of a symplectic cyclic action of a prime order $\geq
5$ is explicitly determined, provided that the action is homologically
nontrivial.
