  Motivated by the recent theoretical study by Okubo $et \ al$ [Phys. Rev.
Lett. ${\bf 108}$, 017206 (2012)] on the possible realization of the
frustration-induced $ symmetric$ skyrmion-lattice state in the $J_1$-$J_2$ (or
$J_1$-$J_3$) triangular-lattice Heisenberg model without the
Dzyaloshinskii-Moriya interaction, we investigate the ordering of the classical
$J_1$-$J_2$ honeycomb-lattice Heisenberg antiferromagnet under magnetic fields
by means of a Monte Carlo simulation, a mean-field analysis and a
low-temperature expansion. The model has been known to have an infinite
ring-like degeneracy in the wavevector space in its ground state for
$1/6<J_2/J_1<0.5$, in distinction with the triangular-lattice model. As
reported by Okumura $et \ al$ [J. Phys. Soc. Jpn. ${\bf 79}$, 114705 (2010)],
such a ring-like degeneracy gives rise to exotic spin liquid states in zero
field, $e.g$, the "ring-liquid" state and the "pancake-liquid" state. In this
paper, we study the in-field ordering properties of the model paying attention
to the possible appearance of exotic multiple-$q$ states. Main focus is made on
the $J_2/J_1=0.3$ case, where we observe a rich variety of multiple-$q$ states
including the single-$q$, double-$q$ and triple-$q$ states. While the
skyrmion-lattice triple-$q$ state observed in the triangular-lattice model is
not realized, we instead observe an exotic double-$q$ state consisting of
meron/antimeron lattice textures.
