  An auto-B\"acklund transformation for the quad equation $\mathrm{Q1}_1$ is
considered as a discrete equation, called $\mathrm{H2}^a$, which is a so called
torqued version of $\mathrm{H2}$. The equations $\mathrm{H2}^a$ and
$\mathrm{Q1}_1$ compose a consistent cube, from which a auto-B\"acklund
transformation and a Lax pair for $\mathrm{H2}^a$ are obtained. More generally
it is shown that auto-B\"acklund transformations admit auto-B\"acklund
transformations. Using the auto-B\"acklund transformation for $\mathrm{H2}^a$
we derive a seed solution and a one-soliton solution. From this solution it is
seen that $\mathrm{H2}^a$ is a semi-autonomous lattice equation, as the spacing
parameter $q$ depends on $m$ but it disappears from the plain wave factor.
