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Consider $i: S^1 \hookrightarrow M_f$ where $f:S^1 \rightarrow S^1$ is squaring and $M_f$ denotes the mapping cylinder. Then the inclusion induces a map $\pi_1(S^1) \rightarrow \pi_1(M_f)$ which has image $2 \mathbb{Z}$. Adding additional segments and attaching disks is the same as adding generators and relations to the presentation $\langle x,y | y=2x \rangle$ and the goal is to end up with $y$ generating the entire group with $|y|=\infty$. This means that we must have $x=ny=2nx$ which implies $x$ has finite order which in turn implies $y$ has finite order. This means that we cannot attach segments and disks to make the inclusion induce an isomorphism.