You are asking for interesting 2-cocycles on super Lie groups which are not just 2-cocycles on the underlying bosonic Lie groups. Here is one example:
Exceptional and fermionic (p+2)-cocycles for p∈N appear on super-Poincaré groups Spin(d−1,1)⋊Rd−1,1|N (where N denotes a choice real spinor representation) for a finite number of triples (d,N,p). In string theory, the table of these nontrivial triples is called the brane scan since there is one such for every spacetime dimension d for supergravity with N-supersymmetries in which super p-branes may propagate.
Hence the 2-cocycles correspond to 0-branes. In particular R9,1|16+¯16 (the type IIA supersymmetry super Lie group) carries such a 2-cocycle, corresponding to the D0-brane in typeII A string theory.
The corresponding projective representations are equivalently ordinary representations of the corresponding central extension. The central extension classified by the D0-brane 2-cocycle on the 10d supertranslation group is curious: it's 11d super-translation group. (This is a super Lie theoretic incarnation of the physics lore that type IIA string theory grows an 11th dimension via D0-brane condensation.)
There is loads of further interesting super Lie theoretic structure hidden in the super Lie cocycles of the brane scan. See The brane bouquet for more.
