Aspace X is said to have the absolutely strongly star -đ€-Hurewicz (ASSđ€H) property if for each sequence (đ°n : n â đ
)of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n â đ
) of finite subsets of Y such that for each x â X, {n â đ
: x â St(Fn, đ°n)}â đ€, where đ€ is the proper admissible ideal of đ
. In this paper, we investigate the relationship between the ASSđ€H property and other related properties and study the topological properties of the ASSđ€H property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASSđ€H properties.