  We develop the basic theory of Maurer-Cartan simplicial sets associated to
(shifted complete) $L_\infty$ algebras equipped with the action of a finite
group. Our main result asserts that the inclusion of the fixed points of this
equivariant simplicial set into the homotopy fixed points is a homotopy
equivalence of Kan complexes, provided the $L_\infty$ algebra is concentrated
in non-negative degrees. As an application, and under certain connectivity
assumptions, we provide rational algebraic models of the fixed and homotopy
fixed points of mapping spaces equipped with the action of a finite group.
