Continuously varying homeomorphisms in a homogeneous space Let $X$ be a topological space, and equip its homeomorphism group $\text{Homeo}(X)$ with the compact-open topology. I probably need to restrict $X$ to be locally compact for it to play off with $\text{Homeo}(X)$ nicely.
The usual definition of $X$ being homogeneous is that given $x,y \in X$ there exists $f \in \text{Homeo(X)}$ with $f(x) = y$. My question: When can this choice of $f$ be made continuous? More precisely, is there a natural class of homogeneous spaces for which there is always a continuous map $$\theta: \{(x,y): x \neq y\} \rightarrow \text{Homeo}(X)$$
such that $\theta(x,y)$ carries $x$ to $y$. Is it true for connected manifolds?

One of such natural classes are topological groups $G=X$ with $\theta(x,y)(t)=yx^{-1}(t)$ for each $t\in G$. Indeed, clearly that $\theta(x,y)(x)=y$ for each $x,y\in G$. To prove the continuity of the map $\theta: G\times G\to \operatorname{Homeo}(X)$ consider a subbase $\mathcal B$ of the space $\operatorname{Homeo}(X)$ consisting of sets $[K,O]=\{f\in\operatorname{Homeo}(X): f(K)\subset O,$ $K$ is a compact subset of $G$  and  $O$ is an open subset of $G \}$.
It suffices to show that $\theta^{-1}(B)$ is open for each $[K,U]\in\mathcal B$. Let $\theta(x,y)\in [K,O]$, that is $yx^{-1}K\subset O$. Continuity of operations of topological group imply that for each $t\in K$ there exists open neighborhoods $U_t$, $V_t$, and $W_t$ of points $y$, $x$, and $t$ respectively such that $U_t^{-1}V_tW_t\subset O$. A family $\{W_t:t\in K\}$ is an open cover of the set $K$. Since the set $K$ is compact, there exists a finite subset $F$ of $K$ such that $K\subset \{W_t:t\in F\}$. Put 
$U=\bigcap  \{U_t:t\in F\}$ and $V=\bigcap  \{V_t:t\in F\}$. Then $U^{-1}VK\subset O$, so $\theta(U,V)\in [K,O]$.
