  We study the ground-state (gs) properties of the frustrated spin-1/2
$J_{1}$--$J_{2}$--$J_{3}$ Heisenberg model on a honeycomb lattice with
ferromagnetic (FM) nearest-neighbor ($J_{1}=-1$) exchange and frustrating
antiferromagnetic (AFM) next-nearest-neighbor ($J_{2}>0$) and
next-next-nearest-neighbor ($J_{3}>0$) exchanges, for the case $J_{3}=J_{2}$.
We use the coupled cluster method in high orders of approximation, complemented
by the exact diagonalization of a lattice with 32 sites, and calculate the gs
energy, magnetic order parameter, and spin-spin correlation functions. We find
a quantum phase transition between regions characterized by FM order and a form
of AFM ("striped") collinear order at $J^{c}_{2} \approx 0.1095 \pm 0.0005$. We
compare results for the FM case (with $J_{1}=-1$) to previous results for the
corresponding AFM case (with $J_{1}=+1$). While the magnetic order parameters
behave similarly for the FM and the AFM models for large values of the
frustration parameter $J_{2}$, there are considerable differences between them
for $J_{2}/|J_{1}| \lesssim 0.6$. For example, the quasiclassical collinear
magnetic long-range order for the AFM model (with $J_{1}=+1$) breaks down at
$J^{c_{2}}_{2} \approx 0.60$, whereas the "equivalent" point for the FM model
(with $J_{1}=-1$) occurs at $J^{c}_{2} \approx 0.11$. Unlike in the AFM model
(with $J_{1}=+1$), where a plaquette valence-bond crystal phase intrudes
between the two corresponding quasiclassical AFM phases (with N\'eel and
striped order) for $J^{c_{1}}_{2} < J_{2} < J^{c_{2}}_{2}$, with $J^{c_{1}}_{2}
\approx 0.47$, we find no clear indications in the FM model for an intermediate
magnetically disordered phase between the phases exhibiting FM and striped
order. Instead, the evidence points strongly to a direct first-order transition
between the two ordered phases of the FM model.
