The bond orders of ${\mathrm{He}}_2^{+}$ and ${\mathrm{He}}_2$ are respectively

$$\begin{gathered} {\mathrm{He}}_2\left(4{\mathrm{e}}^{-}\right)=\mathrm{\sigma }1{\mathrm{s}}^2{\mathrm{\sigma }}^{*}1{\mathrm{s}}^2 \\ \mathrm{Bond} \mathrm{order} =\frac{2-2}{2}=0 \\ {\mathrm{He}}_2^{+}\left(3{\mathrm{e}}^{-}\right)=\mathrm{\sigma }1{\mathrm{s}}^2{\mathrm{\sigma }}^{*}1{\mathrm{s}}^1 \\ \mathrm{Bond} \mathrm{order} =\frac{2-1}{2}=0.5 \end{gathered}$$

Hence , the correct option (A).