An injured cat beside a road needs help quickly, but the first life to protect is yours. A rescuer standing in traffic in the dark helps nobody, and a frightened cat beside moving cars is in more danger with every passing vehicle. Work in order: make the scene safe, contain the cat gently, and get her to professional care. Everything you do is protection and transport; treatment belongs to the vet.
Your Own Safety Comes First
Park safely, well off the carriageway where you can, and switch hazard lights on so other drivers see the scene early. If you have a passenger, let them watch the traffic while you go to the cat; if you are alone, choose your moment and never crouch in a live lane. Wear gloves if you have them, because even a gentle cat in pain may bite hard, and a bite beside a road tempts a sudden movement neither of you can afford.
Approach slowly, from the side away from the traffic, speaking quietly. An injured cat may bite or bolt into traffic if she feels cornered, so give her a second to register you and avoid looming over her. If she drags herself away from you into greater danger, pause rather than grabbing; a few calm seconds often settle her better than a lunge.
Gentle Containment: a Towel or a Box
Use whatever the car offers for gentle containment: a towel, a blanket, a coat, or a sturdy box. Lay the towel over her to dim her view and limit struggling, then lift her supported along her body rather than by any injured part, and settle her into the box or onto a folded blanket on a seat or in the boot area where she cannot slide. If she resists fiercely, drape the towel over her and gather towel and cat together rather than wrestling limb by limb. Keep the container closed or covered for the journey; darkness and confinement calm most cats and stop a sudden escape attempt when a door opens.
Do not offer food or water, do not attempt to straighten, bandage or examine injuries, and do not give any medication. Beyond protection and transport, anything medical can wait for the vet, who is equipped for all of it.
Straight to a Vet
Drive straight to a vet, calling ahead if you can do so safely, or ask a passenger or the practice's out-of-hours line to guide you. Tell them you are bringing an injured stray, where you found her, and roughly what you could see, without exaggerating or diagnosing. Hand over those details in writing if you can: the exact spot matters if an owner comes looking, and the practice may need your account of how she was found. You will not usually be charged for handing in an injured stray, and the practice can scan for a chip once she is stable and begin the search for an owner.
A finder can lean on Lost Pet Alert System through KinsLe as well, sending word to owners close by and opening a direct message route to the owner without publishing personal details.