This is a very vague question.

We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain complexes are modules of $\mathbb{R}^{1|1}$(it is a super Lie group, I shall not call it as dual numbers, since the multiplication here differs from the usual one), more generally, modules of a operad T (Lie algebra and associative algebra are of this kind). To get the infinitified version(L-infinity algebra, A-infinity algebra, etc.) one uses an infinity operad. Note that module is basically a morphism from T to some Auto( ). And the infinitified version basically replace the source thing by a higher thing.

Projective representation goes differently. A representation is the same as a module, a morphism from T to some Auto( ). Projective representation of G, which can be think of as a non-strict functor from BG, the one object category associated with G, to some 2-category. A lot of constructions come from similar ways, they basically replaces the target by a higher thing. Other examples: 2-bundle as a functor Cech groupoid to a 2-category associated to a 2-group, and representation (of a Lie algebra) up to homotopy should also be this kind.

A nature question: whether these two constructions are the same? It seems obvious not. On the other hand, I feel they should have some relations.

To make my question more clear and concrete, consider Lie algebra and L-infinity algebra. Lie algebras are modules of the operad $\mathcal{Lie}$, view as certain map from $\mathcal{Lie}$ to some Auto(). L-infinity algebras are modules of operad $\mathcal{Lie}^\infty$, view as certain map from $\mathcal{Lie}^\infty$ to some Auto(). My question is whether it is possible to view L-infinity algebras as certain map from $\mathcal{Lie}$ to infinitified version of Auto().