For $n \in \mathbf{N}_{+}$, calculate
$$
\begin{array}{l}
\mathrm{C}_{4 n+1}^{1}+\mathrm{C}_{4 n+1}^{5}+\cdots+\mathrm{C}_{4 n+1}^{4 n+1} \\
=
\end{array}
$$
$$
\text { II.7. } 2^{4 n-1}+(-1)^{n} 2^{2 n-1} \text {. }
$$

When $n=1$, $\mathrm{C}_{5}^{1}+\mathrm{C}_{5}^{5}=2^{3}-2$;
When $n=2$, $\mathrm{C}_{9}^{1}+\mathrm{C}_{9}^{5}+\mathrm{C}_{9}^{9}=2^{7}+2^{3}$;
When $n=3$,
$$
\mathrm{C}_{13}^{1}+\mathrm{C}_{13}^{5}+\mathrm{C}_{13}^{9}+\mathrm{C}_{13}^{13}=2^{11}-2^{5} ;
$$

When $n=4$,
$$
\mathrm{C}_{17}^{1}+\mathrm{C}_{17}^{5}+\mathrm{C}_{17}^{9}+\mathrm{C}_{17}^{13}+\mathrm{C}_{17}^{17}=2^{15}+2^{7} \text {. }
$$

From this, we can inductively derive the general conclusion:
$$
\begin{array}{l}
\mathrm{C}_{4 n+1}^{1}+\mathrm{C}_{4 n+1}^{5}+\cdots+\mathrm{C}_{4 n+1}^{4 n+1} \\
=2^{4 n-1}+(-1)^{n} 2^{2 n-1} .
\end{array}
$$