Let $X$ be a topological space and $f:X\to X$ be a homeomorphism. Then the induced map $f_1:\pi_1(X,x)\to\pi_1(X,fx)(\cong \pi_1(X,x))$ is an isomorphism (automorphism up to conjugate). In the following we will adapt this viewpoint and denote $f_1:\pi_1(X)\to\pi_1(X)$.
I think $f_1$ is kind of special if $f$ can be extended to another ambient space. More precisely I want to know:
Let $i:N\to M$ be an embedding of a submaniold $N$ into a closed manifold $M$ such that $i_1(\pi_1(N))=0$. Then for any homeomorphism $f:M\to M$ with $f(N)=N$, what can we say about the restriction $g=f|_N:N\to N$?
In general assume $i:N\to M$ satisfies $\ker(i_1)\neq0$ and $fN=N$. Letting $g=f|_N$, will we have $g_1(\alpha)=\alpha$ for all $\alpha\in\ker(i_1)\le\pi_1(N)$?
See here for more answers with some other interesting examples.
Thank you all!