  We introduce supergroup analogues of 3-manifold invariants $\hat{Z}$, also
known as homological blocks, which were previously considered for ordinary
compact semisimple Lie groups. We focus on superunitary groups, and work out
the case of SU(2|1) in details. Physically these invariants are realized as the
index of BPS states of a system of intersecting fivebranes wrapping a
3-manifold in M-theory. As in the original case, the homological blocks are
q-series with integer coefficients. We provide an explicit algorithm to
calculate these q-series for a class of plumbed 3-manifolds and study quantum
modularity and resurgence properties for some particular 3-manifolds. Finally,
we conjecture a formula relating the $\hat{Z}$ invariants and the quantum
invariants constructed from a non-semisimple category of representation of the
unrolled version of a quantum supergroup.
