Orientation-reversing actions on a solid torus which extend to a lens space

We consider all the finite orientation-reversing actions on a solid torus $V_1$, write explicit representations for each equivalence class of action, and identify the quotient space. For fixed positive integers $n$ and $s$, these groups divide into seven families ($1\leq i\leq 7$) with two groups in each family $\widetilde{G_{(i,j)}}(n,s)$ and ${G_{(i,j)}}(n,s')$, where $s'=s$ or $2s$ depending on certain conditions on $s$ or $n$. The integer $j$ relates to the quotient type $(Bj,n)$. We consider the question of extending these actions to a lens space $L(p,q)=V_1\cup_{\alpha}V_2$. We show $\widetilde{G_{(i,j)}}(n,s)$ extends to the 3-sphere $\mathbb{S}^3$ but not to $L(2,1)$, and ${G_{(i,j)}}(n,s')$ extends to $L(2,1)$ but not to $\mathbb{S}^3$. All the non-orientable orbifolds having an Euler number zero Heegaard decomposition with finite fundamental group appear as quotient spaces of these actions. In addition, no orientation-reversing action on $V_1$ extends to $L(p,q)$ for $p>2$.