Are there some theorems about the splitting of infinite conjugacy classes into several conjugacy classes in a subgroup? I am mainly interested in subgroups of finite index. Thanks.
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I'm not sure what you mean by an infinite length conjugacy class. Most likely you mean that the cardinality is infinite. Consider the Heisenberg group generated by 3 elements $x,y$ and $z$ with relations so that $z$ is central and $xyx^{1}y^{1}=z$. Then the conjugacy class containing $y$ consists of all elements of the form $yz^n$ for integers $n$. If we pass to the finite index subgroup generated by $x^k,y$ and $z$ for some natural number $k$ this splits into $k$ distinct classes represented by $yz^i$ for $i=0,\ldots k1$. 

