= Solution
For the classical subgroup-covering construction, assume $X$ is <path-connected>, <locally path-connected>, and <semilocally simply connected>. These are the usual hypotheses underlying the <classification of connected covering spaces>; an arbitrary space, as literally allowed by the wording, need not admit every <subgroup> covering.
Construct the <path-class construction of a subgroup covering> directly. Consider <paths> $\alpha$ starting at $x_0$, and identify $\alpha,\beta$ when they have the same endpoint and the loop class $[\alpha\beta^{-1}]$ belongs to $H$. The <subgroup> properties make this an equivalence relation. Let $Y$ be its set of classes and define $p([\alpha])=\alpha(1)$.
Choose a path-connected open neighbourhood $U$ of $\alpha(1)$ whose loops become <null-homotopic> in $X$. Such neighbourhoods form a local basis under the stated local hypotheses. Define a sheet
$$
N([\alpha],U)=\{[\alpha\eta]:\eta\text{ is a path in }U\text{ starting at }\alpha(1)\}.
$$
For each point of $U$ the resulting class is independent of the chosen <path> $\eta$, since the difference is a loop in $U$, conjugated by $\alpha$, and is trivial in $\pi_1(X,x_0)$. Thus $p$ maps this sheet bijectively onto $U$. Give $Y$ the topology generated by these sheets. Restrictions to smaller such neighbourhoods show that the sheets form a basis and that their endpoint maps are homeomorphisms. Two sheets over $U$ that intersect coincide, by concatenating and reversing <paths> within $U$; otherwise they are disjoint. Therefore $p^{-1}(U)$ is a disjoint union of copies of $U$, and $p$ is a <covering map>.
The basepoint is the class of the constant <path>. The <path> $\alpha$ lifts to its successive prefix classes, giving a <path> from this basepoint to $[\alpha]$; hence $Y$ is path-connected. A loop $\gamma$ lifts closed exactly when its final prefix class agrees with the constant class, namely when $[\gamma]\in H$. Consequently $p_*(\pi_1(Y))=H$. Part (i) supplies injectivity, so
$$
\boxed{\pi_1(Y)\cong H}.
$$
The missing local hypothesis is substantive. In the <Hawaiian earring>, every neighbourhood of the common basepoint contains a whole sufficiently small circle. Its loop is nontrivial in the whole space, since retracting onto that circle preserves it. An evenly covered neighbourhood would lift that loop to a closed loop in one sheet, so the image <subgroup> of any covering contains a nontrivial element. Therefore the trivial <subgroup> cannot be realized by a simply connected covering of this space. The intended result is valid under the local covering-space hypotheses above, not for every topological space.
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