  We give a partial solution a question of Grigorchuk, Nekrashevych,
Sushchanskii and \v{S}uni\'k by giving an algorithm to test whether a finite
state element of an infinite iterated (permutational) wreath product $\hat G =
\mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr \mathbb Z/k\mathbb Z\wr >...$
of cyclic groups of order $n$ acts spherically transitively. We can also decide
whether two finite state spherically transitive elements of $\hat G$ are
conjugate. For general infinite iterated wreath products, an algorithm is
presented to determine whether two finite state automorphisms have the same
image in the abelianization.
