We study a frustrated spin-$\frac{1}{2}$$J_{1}$--$J_{2}$--$J_{3}$--$J_{1}^{\perp}$ Heisenberg antiferromagnet on an
$AA$-stacked bilayer honeycomb lattice. In each layer we consider
nearest-neighbor (NN), next-nearest-neighbor, and next-next-nearest-neighbor
antiferromagnetic (AFM) exchange couplings $J_{1}$, $J_{2}$, and $J_{3}$,
respectively. The two layers are coupled with an AFM NN exchange coupling
$J_{1}^{\perp}\equiv\delta J_{1}$. The model is studied for arbitrary values of
$\delta$ along the line $J_{3}=J_{2}\equiv\alpha J_{1}$ that includes the most
highly frustrated point at $\alpha=\frac{1}{2}$, where the classical ground
state is macroscopically degenerate. The coupled cluster method is used at high
orders of approximation to calculate the magnetic order parameter and the
triplet spin gap. We are thereby able to give an accurate description of the
quantum phase diagram of the model in the $\alpha\delta$ plane in the window $0
\leq \alpha \leq 1$, $0 \leq \delta \leq 1$. This includes two AFM phases with
N\'eel and striped order, and an intermediate gapped paramagnetic phase that
exhibits various forms of valence-bond crystalline order. We obtain accurate
estimations of the two phase boundaries, $\delta = \delta_{c_{i}}(\alpha)$, or
equivalently, $\alpha = \alpha_{c_{i}}(\delta)$, with $i=1$ (N\'eel) and 2
(striped). The two boundaries exhibit an "avoided crossing" behavior with both
curves being reentrant