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We prove a variant of the well-known Reidemeister-Schreier theorem forfinitely $L$-presented groups. More precisely, we prove that each finite indexsubgroup of a finitely $L$-presented group is itself finitely $L$-presented.Our proof is constructive and it yields a finite $L$-presentation for thesubgroup. We further study conditions on a finite index subgroup of aninvariantly finitely $L$-presented group to be invariantly $L$-presenteditself.

A Reidemeister-Schreier theorem for finitely $L$-presented groups