Fundamental groups induced by inclusion

Question

Let $A,\ \hat A,\ B$ and $\hat B$ be topological groups such that $A\subset \hat A$ and $B\subset \hat B$, where $\hat A,\ \hat B$ are connected and $A,\ B$ are not necessarily connected Suppose that we have the following exact sequences of fundamental groups $$\dots\to \pi_1(A)\to \pi_1(B)\to \mathbb Z^n\to \dots$$ $$\dots\to \pi_1(\hat A)\to \pi_1(\hat B)\to \mathbb Z^m\to \dots$$ Let $m<n$ then is it always true that $\pi_1(\hat A)<\pi_1(A)$?

Answer 1

No, take $A= \{1\}$, $\hat A=S^1$ and for example $B=\hat B=S^1\times S^1$ with $m=1$ and $n=2$ (and zeros at dotted ends).