Here I show how to find $F_{n+2}$, when given $F_{n}$ and $F_{n+4}$ in the Fibonacci sequence.

We can deduce that:

$$F_{n+4} = F_{n+3} + F_{n+2} = (F_{n+2} + F_{n+1}) + F_{n+2} = 2\cdot F_{n+2} + F_{n+1}$$

Since,

$$F_{n+2} = F_{n+1} + F_{n} \implies F_{n+1} = F_{n+2} - F_{n}$$

Therefore, $$F_{n+4} = 2\cdot F_{n+2} + (F_{n+2} - F_{n}) = 3\cdot F_{n+2} - F_{n}$$

$$\implies F_{n+2} = \frac{F_{n+4} + F_{n}}{3}$$