  This article primarily aims at classifying, on certain K3 surfaces, the
elliptic fibrations induced by conic bundles on smooth del Pezzo surfaces. The
key geometric tool employed is the Alexeev-Nikulin correspondence between del
Pezzo surfaces with log-terminal singularities of Gorenstein index two and K3
surfaces with non-symplectic involutions of elliptic type: the latter surfaces
are realized as appropriate double covers obtained from the former ones. The
main application of this correspondence is in the study of linear systems that
induce elliptic fibrations on K3 surfaces admitting a strictly elliptic
non-symplectic involution, i.e., whose fixed locus consists of a single curve
of genus $g\geq 2$. The obtained results are similar to those achieved by
Garbagnati and Salgado for jacobian elliptic fibrations.
