You Do Need power plug adapter while traveling from Cocos (Keeling) Islands to Saint Helena.

Here is Why You Need An Adapter

The outlet and plug type used in Saint Helena is G and in Cocos (Keeling) Islands is I. This is how they look like.

Cocos (Keeling) Islands

In Cocos (Keeling) Islands, outlets of type I are in use.

Type I

Saint Helena

In Saint Helena, outlets of type G are in use.

Type G