  The aim of this paper is to construct "special" isogenies between K3
surfaces, which are not Galois covers between K3 surfaces, but are obtained by
composing cyclic Galois covers, induced by quotients by symplectic
automorphisms. We determine the families of K3 surfaces for which this
construction is possible. To this purpose we will prove that there are
infinitely many big families of K3 surfaces which both admit a finite
symplectic automorphism and are (desingularizations of) quotients of other K3
surfaces by a symplectic automorphism.
  In the case of involutions, for any $n\in\mathbb{N}_{>0}$ we determine the
transcendental lattices of the K3 surfaces which are $2^n:1$ isogenous (by a
non Galois cover) to other K3 surfaces. We also study the Galois closure of the
$2^2:1$ isogenies and we describe the explicit geometry on an example.

Algebraic Geometry