The concept of an automatic group can be generalized to a group that is aut
omatic with respect to a specified subgroup. This means that there is a fin
ite state automaton that recognizes a unique word in each coset of the subg
roup, and others that essentially recognize the permutation action on these
cosets induced by multiplying by a group generator. These automata make it
possible to enumerate coset representatives as words in the generators, an
d to solve the generalized word problem for the subgroup efficiently. Algor
ithms to construct these automata have been described previously by Redfern
. Here we describe improved versions, together with implementation details
and some examples of successful calculations. A related algorithm to comput
e a finite presentation of the subgroup is also described. (C) 1999 Academi
c Press.