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

Integrability of auto-B\"acklund transformations,and solutions of a torqued ABS equation