  In this note, we compute the order and provide the structure of the unit
group $\mathcal{U}(FD_{2p^m})$ of the group algebra $FD_{2p^m}$, where $F$ is a
finite field of characteristic 2 and $D_{2p^m}$ is the dihedral group of order
$2p^m$ such that $p$ is an odd prime. Further, we obtain the structure of the
unitary subgroup $\mathcal{U}_*(FD_{2p^m})$ with respect to canonical
involution * and prove that it is a normal subgroup of the unit group
$\mathcal{U}(FD_{2p^m})$.
