The K3 surface are one of the two classes of Kähler surfaces which are endowed with a symplectic structure. An automorphism α on a K3 surface X is called symplectic if it preserves this structure and if it has finite order, the desingularization of the quotient X/α is still a K3 surface, a priori different from X.So, one constructs a relation (the quotient by an automorphism) between different families of K3 surfaces. The families of projective K3 surfaces admitting a symplectic involution and the ones of their quotient are intensively studied in the last decades. In this talk, I will present the classical known results for the generic member of these families and then I consider specializations of some of them. In particular, I discuss the cases in which the K3 surface admitting the involution and its quotient are contained in the same family, and/or are isomorphic. The first and most famous example of this phenomenon is the one in which the symplectic involution is induced by a translation by a 2-torsion section on an elliptic fibration, i.e. it is a van Geemen--Sarti involution. We provide other examples and study specializations of both van Geemen--Sarti involutions and of the other ones presented.