Z_p^3 × Z_q is a CI-group w.r.t. binary relational structures for distinct primes p and q.
Somlai, Elementary abelian p-groups of rank 2 p + 3 are not CI-groups, Journal of Algebraic Combinatorics, 34(3) (2011), 323-335
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The Cayley isomorphism property for $\mathbb{Z}_p^3 \times \mathbb{Z}_q$
Z_p^3 × Z_q is a CI-group w.r.t. binary relational structures for distinct primes p and q.