First a way to make things more clear.

Keep throwing a fair die again and again resulting in an endless string like $$32464344162634521466326...$$

Now have a look at the substrings that end with a $6$, do not contain another $6$, contain only even digits and are of maximal length.

That gives in my example the substrings $246$, $6$, $26$, $46$, $6$, $26$.

Now the question can be interpreted as:

What is the average length of substrings like these?

Finding this comes to the same as finding the expectation of the number of throws needed to arrive at $1,3,5$ or $6$.

If we denote that expectation by $\mu$ then we have the equality:$$\mu=\frac23\cdot1+\frac13(1+\mu)=1+\frac13\mu$$or equivalently: $$\mu=\frac32$$

P.S.

When I saw this problem for the first time (in a question that is now marked as a duplicate of this question) I was completely wrong in my thinking and the answer based on my intuition was $3$.