Given a (short exact) sequence of abelian groups. Are the following true?
Case 1: $0\to A\to B\to B/A\to 0$ is exact and splits for all $A\leq B$.
Case 2: If $0\to A\to B\to C\to 0$ is exact and B is free abelian then the sequence it splits.
Case 3,4: Similar as in 2, but saying $A,C$ are free abelian respectively.
The original question was: a short exact sequence $0\to A\to\mathbb{Z}^8\to\mathbb{Z}\to0$ implies $A\cong\mathbb{Z}^7$?
Assertions in cases 1 2 and 3 are false. Counter-example for all: $$0\to n\mathbf Z\to\mathbf Z\to \mathbf Z/n \mathbf Z\to 0 $$ doesn't split since $ \mathbf Z/n \mathbf Z$ has torsion and a direct summand of $ \mathbf Z$ doesn't.
On the contrary, if $C$ is free, the exact sequence splits, since it's enough to take as images of the elements in a basis of $C$ any element in their inverse images.