To better understand the fundamental quandle of a knot or link, it can be useful to look at finite quotients of the quandle. One such quotient is the $n$-quandle (or, when $n=2$, the {em involutory} quandle). Hoste and Shanahan cite{HS2} gave a complete list of the links which have finite $n$-quandles; it remained to give explicit descriptions of these quandles. This has been done for several cases in cite{CHMS} and cite{HS1}; in the current work we continue this project and explicitly describe the Cayley graphs for the finite involutory quandles of two-bridge links with an axis.
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