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 V₁, 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≤ i≤ 7) with two groups in each family 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)=V₁∪αV₂. We show G(i,j)(n,s) extends to the 3-sphere S³ but not to $L(2,1)$, and G(i,j)(n,s') extends to $L(2,1)$ but not to S³. 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 V₁ extends to $L(p,q)$ for $p>2$.