CI-groups with respect to ternary relational structures: new examples

We find a sufficient condition to establish that certain abelian groups are not CI-groups with respect to ternary relational structures, and then show that the groups $\Z_3\times\Z_2^2$, $\Z_7\times\Z_2^3$, and $\Z_5\times\Z_2^4$ satisfy this condition. Then we completely determine which groups $\Z_2^3\times\Z_p$, $p$ a prime, are CI-groups with respect to binary and ternary relational structures. Finally, we show that $\Z_2^5$ is not a CI-group with respect to ternary relational structures.