Let $G$ be a finitely generating set and $H\leq G$ be a subgroup with finite index, this means that there is finite set $\{g_i\}_{i=1}^n\subseteq G$ such that $G= \bigcup_{i=1}^n g_iH$. Let $A$ be a symmetric finitely generating set of $H$. We can add more elements to $A$ to get a symmetric finitely generating set $S$ of $G$.
A map $F:G\to X$ is called a $\delta$- pseudo orbit for continuous action $\varphi:G\times (X, d)\to (X, d)$ on metric space $(X, d)$, if $d(F(sg),\varphi(s, F(g)))<\delta$ for all $s\in S$ and all $g\in G$.
In my research, I have a $\delta$-pseudo orbit $f:H\to X$ for $\varphi:H\times X\to X$ i.e. $d(f(ah), \varphi(a, f(h))<\delta$ for all $a\in A$ and all $h\in H$, where $H=\langle A\rangle$.
I need to find a pseudo orbit $F:G\to X$ for $\varphi:G\times X\to X$ such that $F(h)= f(h)$ for all $h\in H$. Can you help me to know such extension of pseudo orbit is true or not?