  We use the coupled cluster method (CCM) to study a frustrated
spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{1}^{\perp}$ Heisenberg antiferromagnet
on a bilayer honeycomb lattice with $AA$ stacking. Both nearest-neighbor (NN)
and frustrating next-nearest-neighbor antiferromagnetic (AFM) exchange
interactions are present in each layer, with respective exchange coupling
constants $J_{1}>0$ and $J_{2} \equiv \kappa J_{1} > 0$. The two layers are
coupled with NN AFM exchanges with coupling strength $J_{1}^{\perp}\equiv
\delta J_{1}>0$. We calculate to high orders of approximation within the CCM
the zero-field transverse magnetic susceptibility $\chi$ in the N\'eel phase.
We thus obtain an accurate estimate of the full boundary of the N\'eel phase in
the $\kappa\delta$ plane for the zero-temperature quantum phase diagram. We
demonstrate explicitly that the phase boundary derived from $\chi$ is fully
consistent with that obtained from the vanishing of the N\'eel magnetic order
parameter. We thus conclude that at all points along the N\'eel phase boundary
quasiclassical magnetic order gives way to a nonclassical paramagnetic phase
with a nonzero energy gap. The N\'eel phase boundary exhibits a marked
reentrant behavior, which we discuss in detail.
