= Solution
A <Serre fibration> $p:E\to B$ has the <homotopy lifting property> for every disc: given $h:D^k\times I\to B$ and $\widetilde h_0:D^k\to E$ with $p\widetilde h_0=h(-,0)$, there is a lift $\widetilde h$ extending $\widetilde h_0$. Equivalently one may test this property on <CW complexes>.
For a map $g:E\to B$, its <mapping-path fibration replacement> is
$$
E_g=\{(e,\gamma):\gamma(0)=g(e)\},\qquad
p(e,\gamma)=\gamma(1).
$$
Path spaces have the compact-open topology, with the usual compactly generated convention. The inclusion $s(e)=(e,\operatorname{const}_{g(e)})$ satisfies $ps=g$. Forgetting the path is a <homotopy inverse> to $s$: shorten $\gamma$ to $u\mapsto\gamma(tu)$, from its constant initial path at $t=0$ to the original path at $t=1$. The equivalence here means a commuting triangle over the same base whose map of total spaces is a <homotopy equivalence>; it does not assert a fibrewise homotopy inverse over the base. Appending the successive segments of a moving base path gives the <homotopy lifting property> for $p$, even for arbitrary parameter spaces, so $p$ is in particular a <Serre fibration>. Its ordinary fibre over $b_0$ is the <homotopy fibre>
$$
\operatorname{hofib}_{b_0}(g)=\{(e,\gamma):\gamma(0)=g(e),\ \gamma(1)=b_0\}.
$$
For a based <Serre fibration>, with fibre inclusion $j:F\hookrightarrow E$, the <long exact sequence of homotopy groups of a fibration> is
$$
\cdots\longrightarrow\pi_k(F)\xrightarrow{j_*}\pi_k(E)
\xrightarrow{p_*}\pi_k(B)\xrightarrow{\partial}\pi_{k-1}(F)
\longrightarrow\cdots\longrightarrow\pi_1(B)\xrightarrow{\partial}\pi_0(F)
\longrightarrow\pi_0(E)\longrightarrow\pi_0(B).
$$
The maps $j_*,p_*$ compose based sphere representatives with $j,p$; the maps on <path components> do the same with points. To define $\partial$, represent a class by $(I^k,\partial I^k)\to(B,b_0)$ and lift the cube starting at $e_0$ on its bottom face, keeping its vertical sides at $e_0$. The opposite face then gives $(I^{k-1},\partial I^{k-1})\to(F,e_0)$. Relative lifting for a <CW pair> gives this construction; its based homotopy class is independent of choices. At $k=1$ it records the component of the endpoint of the lifted loop. The last terms are pointed sets rather than groups in general.
Now write $W=S^1\vee S^n$ and let $f:W\to S^n$ be the collapse map. Inclusion of the $n$-sphere is a section, so $f_*:\pi_k(W)\to\pi_k(S^n)$ is surjective in every positive degree. Thus all connecting maps in its <homotopy fibre> sequence vanish. The fibre is <path-connected>, and
$$
\pi_1(F)\cong\pi_1(W)\cong\mathbb Z,
\qquad
\pi_n(F)\cong\ker\bigl(\pi_n(W)\to\pi_n(S^n)\bigr).
$$
The section also gives the first assertion when $n=2$, by killing the connecting map from $\pi_2(S^2)$.
The <universal cover> of $W$ consists of a line with an $n$-sphere attached at each integer. It is homotopy equivalent to the <wedge sum> $\bigvee_{k\in\mathbb Z}S^n$. This cover is $(n-1)$-connected, so the <Hurewicz theorem> identifies its $n$th <homotopy group> with its $n$th <homology>. Choose a generator $t$ of the <fundamental group> whose action shifts the sphere indexed by $k$ to that indexed by $k+1$. The resulting <homotopy groups as modules over the fundamental group> give
$$
\pi_n(W)\cong R=\mathbb Z[t,t^{-1}].
$$
Every lifted sphere maps to $S^n$ with degree one. Consequently $f_*$ is the augmentation $\varepsilon(\sum a_kt^k)=\sum a_k$, with the trivial $R$-action on $\mathbb Z$. The inclusion of the fibre respects the <fundamental group> actions, so
$$
\boxed{\pi_n(F)\cong\ker\varepsilon=(t-1)R\cong R
\quad\text{as a }\mathbb Z[\pi_1(F)]\text{-module}}.
$$
The <augmentation ideal of the infinite cyclic group> is generated by $t-1$: multiply a Laurent polynomial by a power of $t$ and use divisibility by $t-1$ when its value at one vanishes. Multiplication by $t-1$ is injective, giving the last isomorphism. This is a free module of rank one, whose underlying <abelian group> has countably infinite rank; its $\pi_1$-action is not trivial.
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