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A modification of Gumtram's example could produce a countable group with the required property, which is not an FC-group. Let $G$ be the direct product of non-abelian symmetric groups $G=\times_{n\ge 3} S_n$, where $S_n$ is the symmetric group on $n$ elements. For each $n$, $S_n$ has an inner automorphism $\alpha_n$, of degree $n$ (conjugation by an $n$-cycle). Now consider the semidirect product $H=G \rtimes \langle a \rangle$, of $G$ with an infinite cyclic group $\langle a \rangle$, such that $a$ preserves every direct factor $S_n$ of $G$, acting on it by $\alpha_n$. I think that the group $H$ satisfies Drike's property, but is not and FC-group. The centralizer of $a$ will have infinite index and the conjugacy class of $a$ will be infinite.