Existence of a certain homeomorphism Let $A$ be a compact topological space, and $X$ and $Y$ be homeomorphic Hausdorff spaces. Let $i_1$ and $i_2$ be continuous injections of $A$ into $X$ and $Y$ respectively. Does there exists a homeomorphism $g: X \to Y$ such that $i_2 = g \circ i_1$? If this is not true, what additional properties would be required to make this true?
My attempt at showing there exists such a $g$ didn't really lead anywhere, I think. I first tried showing that given two homeomoprhic compact subspaces of $X$ and $Y$, there exists a homeomorphisms $f$ from $X$ to $Y$ sending one subspace to the other. This I don't know if it is true. Assuming this, I took $i_1 (A)$ and $i_2 (A)$ to be these two subspaces, and I then tried to show there is an automorphism $h$ of $Y$ fixing $i_2 (A)$ such that $h \circ f \circ i_1 = i_2$, and finally taking $g=h \circ f$. This I don't know either if it's true.
Is this the correct approach so solving this problem?
 A: For a trivial counterexample, let $A=\{a\}$ be the one-point space and $X=Y=[0,2]$. Let $i_1(a)=0$ and $i_2(a)=1$. There is no autohomeomorphism of $[0,2]$ sending $0$ to $1$, since $1$ is a cut point, and $0$ is not.
A little less trivially, $A$ could be the subspace $\{0\}\cup\{2^{-n}:n\in\Bbb N\}$ of $\Bbb R$, with $X$ and $Y$ as before. Take $i_1(x)=x$ for $a\in A$ and $i_2(x)=x+1$, and you have the same problem as in the first example. As long as $X$ has points $p$ and $q$ such that no autohomeomorphism of $X$ sends $p$ to $q$, you can construct such a counterexample.
Finally, let
$$X=\big([-2,2]\times\{0\}\big)\cup\big(\{0\}\times[-2,2]\big)\,;$$
this is a nice compact, connected metric space. Let $A=[0,1]$. Let
$$i_1:A\to X:x\mapsto\langle x,0\rangle$$
and
$$i_2:A\to X:x\mapsto\langle x+1,0\rangle\,.$$
Both maps are embeddings of $A$ into $X$, but there is no autohomeomorphism of $X$ mapping the origin to $\langle 1,0\rangle$: deleting the origin from $X$ leaves $4$ components, while deleting $\langle 1,0\rangle$ leaves only $2$.
