does s.e.s 0->A->B->C->0 of profinite groups imply C=B/A and A

does s.e.s 0->A->B->C->0 of profinite groups imply C=B/A and A
2013-01-03T21:57:15Z

Assume $A, B, C$ are profinite groups and $0\to A\to B\to C\to 0$ is an exact sequence of continuous maps. Which of the following assertions follows?:

(i) the subspace-topology induced on $A$ via $A\hookrightarrow B$ agrees with the given one.

(ii) the quotient-topology induced on $C$ via $B\twoheadrightarrow C$ agrees with the given one.