Every abelian group X oi finite rank arises as the middle group of an extension e: 0 -> F -> A" -> T->0 where F is free of finite rank zz and T is torsion with the p-ranks of T finite for all primes p. Given such a T and F we study the equivalence classes of such extensions which result from stipulating that two extensions e . : 0 -» F -> X. -» T -> 0, i = 1, 2, are equivalent if e. = /3e_a for a. e Aut(T) and s e Aut(F). We reduce the problem to T p-primary of finite rank, where in the one case T is injective, and in the other case T is reduced. Suppose TII • _, T.. In our main theorems we prove that in each case these equivalence classes of extensions are in 1-1 correspondence with the equivalence classes of n-generated subgroups of E where £=11TM, E ., E. = End(7\). Two zz-generated subgroups of E will be called equivalent if one can be mapped onto the other by an automorphism of E. 1. With few exceptions our notation will be that of [lj. One of the difficulties in studying groups of extensions Ext(A, B) is that the same group may appear as a middle group in two distinct elements of Ext(A, B). With this in mind we consider the action of the rings End (A) and End(B) on Ext(A, B). In the problem at hand we are interested in the equivalence classes of Ext(T, F) induced by the action of automorphisms of T and F where T is a torsion group with its p-ranks finite for all primes p and F a free abelian group of rank n. Since Ext (T, F) = U Ext (T , F), where T is the p-primary component of T, one may restrict himself to the case when T is p-primary. (See the remark following Theorem 1.2.) Since T is the direct sum of an injective group of finite rank and one which is finite we may treat each case separately. We now state our main theorems. Theorem 1.1. Let T be an injective p-primary group of rank m. Then the equivalence classes of extensions of the form 0 —► F —> X —> T —► 0 are in 1-1 correspondence with the equivalence classes of n-generated subgroups of E. Presented to the Society, January 17, 1974; received by the editors October 5, 1973 and, in revised form, May 3, 1974. AMS (MOS) subject classifications (1970). Primary 20K35, 20J05.