Ongoing Misery

[kadance]

Sounds like you don't get the extension except when you do the requisite 'Cast or Sustain a hex'

[Dr. Frank Funkelstein]

No action for the familiar, but you have to fulfill the 'cast or sustain a hex' condition.

[Ravingdork]

There's no action requirement for either you or your familiar to use Ongoing Misery, save for the action or actions you must use to cast or sustain a hex in order to trigger it.

Ongoing Misery wrote:

When you Cast or Sustain a hex, your familiar can curse a creature within 15 feet of it, prolonging the duration of any negative conditions affecting it by 1 round.

So, in short, you would need to use either a Cast a Spell action or the Sustain action. In either case you only trigger the familiar ability if it is a hex spell that you are casting or maintaining, your familiar has a target within range, and that target is suffering from a valid condition that can be impacted by Ongoing Misery.

I hope that helps!