Topological distinguishibilty of $\infty$ after one point compactification?

Question

Let $X$ be the one point compactification of some locally compact Hausdorff space. Let $\infty \in X$ represent the added point.

Is there always a homomorphism $\phi:X \to X$ with $\phi: \infty \mapsto x \ne \infty$?

In other words: can you always find a homomorphism from $X$ to itself that maps the point $\infty$ to a different point in X?

It seems to me the answer is supposed to be no.

Answer 1

Take $\{\frac{1}{n}\,|\,n\in\mathbb{N}\}$ and compactify it with $0$. Any homeomorphism takes the open set $\{\frac{1}{n}\}$ to an open set, i.e. not to $\{0\}$, so $\{0\}$ is fixed.

Answer 2

Consider the space that is a circle and two of its diameters minus their intersection (the center of the circle). The one point compactification of this space just adds the center of the circle: $$\Huge{\otimes}$$ but there are no other points with $$\Huge{\times}$$ shaped neighborhoods in this space, so every self-homeomorphism of the one point compactification sends the point at infinity to itself.