This is the same as the induced map $\pi_4(KO) \to \pi_4(KU)$. Via Bott periodicity (see also this answer), this is the same as the induced map $\pi_0(KSp) \to \pi_0(KU)$ coming from the map sending a stable quaternionic vector bundle on a point to its underlying stable complex vector bundle relative to some choice of embedding $\mathbb{C} \to \mathbb{H}$. This map doubles dimensions.

This is the same as the induced map $\pi_4(KO) \to \pi_4(KU)$. Via Bott periodicity (see also this answer), this is the same as the induced map $\pi_0(KSp) \to \pi_0(KU)$ coming from the map sending a stable quaternionic vector bundle on a point to its underlying stable complex vector bundle relative to some choice of embedding $\mathbb{C} \to \mathbb{H}$. This map doubles dimensions.