C.E. Aull \cite{aull} defined that a subspace $Y$ of a space $X$ 
is $\alpha$-countably paracompact in $X$ 
if for every countable collection $\mathcal{U}$ of open subsets of $X$ 
with $Y\subset \bigcup \mathcal{U}$,\ 
there exists a collection $\mathcal{V}$ of open subsets of $X$ 
such that $Y\subset \bigcup \mathcal{V}$,\ 
$\mathcal{V}$ is a partial refinement of $\mathcal{U}$ 
and $\mathcal{V}$ is locally finite in $X$.\ 
In this paper,\ we prove that a Tychonoff space $Y$ 
is $\alpha$-countably paracompact in every larger Tychonoff space 
if and only if $Y$ is countably compact.\ Moreover,\ 
by introducing some notions of relative expandability,\ 
we develop our study to characterize their absolute embeddings.\ 
Finally,\ a negative answer to the question 
on relative discrete expandability in \cite{ggmt} is also given.