Let $\mathcal{L}_1$ be the language over alphabet $\{0,1\}^*$. Define language $\mathcal{L}_2$, call even-$\mathcal{L}_1$, as:
$$\mathcal{L}_2 = \{ w_2 w_4 \ldots w_{k} ~:~ w_1 w_2 w_3 w_4 \ldots w_{k-1} w_{k} \in \mathcal{L}_1, ~ \text{k even} \}$$
I would like to construct a DFA or NFA for $\mathcal{L}_2$, but I am having trouble visualizing this computation.
One idea for constructing an automata for $\mathcal{L}_2$ is to suppose that $\mathcal{L_1}$ is composed of two languages:
$$\mathcal{L}_2 = \{ w_2 w_4 \ldots w_{k} ~:~ w_1 w_2 w_3 w_4 \ldots w_{k-1} w_{k} \in \mathcal{L}_1, ~ \text{k even} \}\\ \mathcal{L}_3 = \{ w_1 w_3 \ldots w_{k-1} ~:~ w_1 w_2 w_3 w_4 \ldots w_{k-1} w_{k} \in \mathcal{L}_1, ~ \text{k-1 odd} \}.$$
So that $$\mathcal{L}_1 = \mathcal{L}_2 ~\text{`interleave`}~ \mathcal{L}_3.$$ But then I need to define interleave in terms the automata it operates over, and prove that interleave takes two regular languages and construct another regular language, and since $\mathcal{L}_1$ is regular by assumption, then $\mathcal{L}_2$ and $\mathcal{L}_3$ must both be regular.
Hint: Suppose you have an automata for $\mathcal L_1$. Duplicate its set of states and send the letters between them in both directions.