  We define a $\mathbb{Z}_2$-valued index for stably short-range entangled
states of two-dimensional fermionic lattice systems with charge conservation
and time reversal symmetry. The index takes its non-trivial value precisely if
the `fluxon', the state obtained by inserting a $\pi$-flux through the system,
transforms under time reversal as part of a Kramers pair. This index extends
the Fu-Kane-Mele index of free fermions to interacting systems.
