When you have a spirit summoned, but it's not actually doing anything, it sits in astral space on standby, taking no actions. It takes a Simple Action to call a previously-summoned spirit (this makes it appear in the summoner's vicinity in astral space), then a second Simple Action to issue a command to a group of spirits.
As soon as it is called in combat, a spirit is entitled to roll its initiative; astral spirits all use (Force × 2) + 3D6. Since the Combat Turn is already underway, the spirit will need to subtract 10 from the result for each Initiative Pass that has already gone by. Note that while spirits have Edge equal to half their Force, summoned and bound spirits do not (SR5, p. 304); however, the summoner can use his own Edge on his spirits' tests, and they can use that to Blitz.
When the spirit is ordered into combat, what it does depends on its target. If the target is in astral space or is dual-natured, it can attack without materializing first. In this situation, the spirit must use Mana-type powers (like Confusion or Fear).
If the target is only in the physical world, or the spirit wants to use Physical powers (like Accident or Elemental Attack), it must first take a Complex Action to use the Materialization power. When it does, this changes its physical attributes to the numbers listed for its type (for example, a Force 4 air spirit changes its Reaction from 4 to 8), and its Initiative changes as well.
Per the rules on Changing Initiative (SR5 p. 160), you apply the difference between the base Initiative scores, then add/subtract any changes in dice. To use the Force 4 air spirit as an example, its astral initiative was 8 + 3D6; its physical initiative is 12 + 2D6, so you would add 4 and then subtract 1D6.
To revisit your example:
- Initiative 26: The mage calls his spirit (let's say its that Force 4 air spirit who rolls, say, 19), then orders it into combat.
- Initiative 20: The mook acts.
- Initiative 19: The spirit materializes. It rolls 1D6, getting a 3, so its initiative goes up by 1. Its next action will be on 10 instead of 9.
- Initiative 16: The mage acts.
- Initiative 10: The spirit wins the tie (because its Reaction is now 8), so it attacks the mook. The mook then acts, if he's able.
- Initiative 6: The mage acts.