  Let $H$ be a subgroup of $\pi_{1}(X,x_{0})$. In this paper, we extend the
concept of $X$ being SLT space to $H$-SLT space at $x_0$. First, we show that
the fibers of the endpoint projection $p_{H}:\tilde{X}_{H}\rightarrow X$ are
topological group when $X$ is $H$-SLT space at $x_0$ and $H$ is a normal
subgroup. Also, we show that under these conditions the concepts of
homotopically path Hausdorff relative to $H$ and homotopically Hausdorff
relative to $H$ coincide. Moreover, among other things, we show that the
endpoint projection map $p_{H}$ has the unique path lifting property if and
only if $H$ is a closed normal subgroup of $\pi_{1}^{qtop}(X,x_{0})$ when $X$
is SLT at $x_{0}$. Second, we present conditions under which the whisker
topology is agree with the quotient of compact-open topology on
$\tilde{X}_{H}$. Also, we study the relationship between open subsets of
$\pi_{1}^{wh}(X,x_{0})$ and $\pi_{1}^{qtop}(X,x_{0})$.
