We construct two practical algorithms for twisted conjugacy classes of polycyclic-by-finite groups. The first algorithm determines whether two elements of a group are twisted conjugate for two given endomorphisms, under the condition that the Reidemeister coincidence number of these endomorphisms is finite. The second algorithm determines representatives of the Reidemeister coincidence classes of two endomorphisms if their Reidemeister coincidence number is finite, or returns "fail" if the Reidemeister coincidence number is infinite.

中文翻译：

---

多环有限群的扭曲共轭类算法

我们为多环有限群的扭曲共轭类构建了两种实用算法。在这些自同态的 Reidemeister 重合数是有限的条件下，第一种算法确定群的两个元素对于两个给定的自同态是否是扭曲共轭的。如果 Reidemeister 巧合数有限，则第二种算法确定两个自同态的 Reidemeister 巧合类的代表，如果 Reidemeister 巧合数是无限的，则返回“fail”。