  We analyse the geometry of a spherically symmetric black hole with an inner
and outer apparent horizon which is perturbed by spherical null shells of
matter. On a classical level we observe that the mass inflation instability is
triggered, resulting in a growth of curvature and an inward displacement of the
inner horizon. We study in detail the inner structure of the mass-inflated
region and compare it with previous results obtained for the case in which the
perturbing matter content has a continuous distribution. We then perform an
approximate calculation of the renormalised stress-energy tensor of a quantum
field in the vicinity of the inner horizon, and analyse the semiclassical
backreaction on this region of the geometry. We find that the classical
tendency for this horizon to move inward due to mass inflation is challenged
and potentially overcome by a semiclassical tendency for it to inflate outward.
