kitchen cooker hood through tiled vent outlet

mutley74

The house i moved into has a cooker extract unit fit which currently extracts and vents through the adjoining wall into the garage.
The garage roof has been extended by previous vendors into a tiled (about 45 degree angled) roof.

I would like to vent the cooker hood externally instead of vapours/odour going to garage. Going through the garage external wall is not an easy option, as any hose/duct will require supports (which maybe more supports?) to avoid head height, although horizontal distance is 2.8m.

Vertical distance to tile roof from cooker hood extract outlet is approx. 1.2m.

What sort of tiled vent is suitable for a cooker hood, or is it better to try support the extract hose to through the external garage wall?

I have had builder quotes for a venting through the roof from £170 to £250. A handyman quoted £100 inc but he i did not know him or quality of work.

any advice appreciated.