Let $G=\langle S\rangle$ be a finitely generated group and $\varphi:G\times X\to X$ be a continuous action of $G$ on metric space $(X, d)$. The map $F:G\to X$ is called $\delta$-pseudo orbit for $\varphi:G\times X\to X$, if $d(F(sg), \varphi(s, F(g))<\delta$ for all $s\in S$ and every $g\in G$. For example let $\varphi:\mathbb{Z}\times X\to X$ define by $\varphi(n, x)=f^n(x)$. Then $F:\mathbb{Z}\to X$ with $F(n)=x_n$, is $\delta$-pseudo orbit for $\varphi:\mathbb{Z}\times X\to X$ if $d(f(x_n), x_{n+1})=d(F(n+1), \varphi(1, F(n))<\delta$ and $d(f^{-1}(x_n), x_{n-1})d(F(n-1), \varphi(-1, F(n))<\delta$ for all $n\in\mathbb{Z}$.
It is easy to see that if $\{x_n\}$ is $\delta$-pseudo orbit for $f^2:X\to X$, then $\{y_n\}$ define by $y_{2n}=x_n, y_{2n+1}=f(x_n)$ is $\delta$-pseudo orbit for $f:X\to X$.
Note that $2\mathbb{Z}$ is finite index subgroup of $\mathbb{Z}$. Indeed Subgroup $H=\langle A \rangle$ of $G=\langle S\rangle$ with $A\subseteq S$ is finite index subgroup of $G$ if there is finite set $K\subseteq G$ with $G=KH$.
In my research $H$ is a finite index subgroup of $G$, I have a $\delta$-pseudo orbit $F:H\to X$ for $\varphi:H\times X\to X$ and I need to extend it to a $\delta$-pseudo orbit $F_1:G\to X$ of $\varphi:G\times X\to X$ such that $F(H)\subseteq F_1(G)$.
Would you please help me to know it.