One-point extensions of locally compact paracompact spaces

A space $Y$ is called an {\em extension} of a space $X$ if $Y$ contains $X$ as a dense subspace. Two extensions of $X$ are said to be {\em equivalent} if there is a homeomorphism between them which fixes $X$ point-wise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous function of $Y'$ into $Y$ which fixes $X$ point-wise. An extension $Y$ of $X$ is called a {\em one-point extension} if $Y\backslash X$ is a singleton. An extension $Y$ of $X$ is called {\em first-countable} if $Y$ is first-countable at points of $Y\backslash X$. Let ${\mathcal P}$ be a topological property. An extension $Y$ of $X$ is called a {\em ${\mathcal P}$-extension} if it has ${\mathcal P}$. In this article, for a given locally compact paracompact space $X$, we consider the two classes of one-point \v{C}ech-complete ${\mathcal P}$-extensions of $X$ and one-point first-countable locally-${\mathcal P}$ extensions of $X$, and we study their order-structures, by relating them to the topology of a certain subspace of the outgrowth $\beta X\backslash X$. Here ${\mathcal P}$ is subject to some requirements and include $\sigma$-compactness and the Lindel\"{o}f property as special cases.