  The memory effect at null infinity, $\mathcal{I}^+$, can be defined in terms
of the permanent relative displacement of test particles (at leading order in
$1/r$) resulting from the passage of a burst of gravitational radiation. In
$D=4$ spacetime dimensions, the memory effect can be characterized by the
supertranslation relating the "good cuts" of $\mathcal{I}^+$ in the stationary
eras at early and late retarded times. It also can be characterized in terms of
charges and fluxes associated with supertranslations. Black hole event horizons
are in many ways analogous to $\mathcal{I}^+$. We consider here analogous
definitions of memory for a black hole, assuming that the black hole is
approximately stationary at early and late advanced times, so that its event
horizon is described by a Killing horizon (assumed nonextremal) at early and
late times. We give prescriptions for defining preferred foliations of
nonextremal Killing horizons. We give a definition of the memory tensor for a
black hole in terms of the "permanent relative displacement" of the null
geodesic generators of the event horizon between the early and late time
stationary eras. We show that preferred foliations of the event horizon in the
early and late time eras are related by a Chandrasekaran-Flanagan-Prabhu (CFP)
supertranslation. However, we find that the memory tensor for a black hole
horizon does not appear to be related to the CFP symmetries or their charges
and fluxes in a manner similar to that occurring at $\mathcal{I}^+$.
