We recently purchased a cheap I akaso go pro like camera from Amazon for my wife to attach to her riding helmet when goes horse riding. It's only for fun and we weren't expecting much from this cam but it's awful beyond belief, we only wanted 1080p and it makes a mess of that and yet it's supposed to be 4k. It's going back obviously. I'm thinking the best idea is to get a 2nd hand actual go pro from cex, are there any models we should maybe avoid, looking at a budget of sub £100.
Unfortunately cheap camera are cheap for a reason but asaka do some good models just depends on which you bought. You probably want a cam with good stability and light balance, also one with good wind noise cut out so I'd look at a proper go pro and probably no less than a 5 As an alternative the Sony x1000 cam be picked up for < £100, it lacks go pro accessories (uses a standard camera mount) but the stabilization and audio is excellent, I still use a couple of these, no good at 4k as there's no stabilisation at 4k but 1080p steady shot is great. Go pro has better accessories and cheap batteries.
I'm looking at the Olfi one.five camera at the moment. Quite a bit over your budget but the review was excellent. I can't track the review down at the moment as I think it was in one of the numerous motorcycle magazines I'm reading.
It doesn't have image stabilization so it's all down to how good the software is at tuning this out, I don't know how important that is for horse riding but can't imagine it is super smooth?
Managed to bag an HD version for £50 on CEX, was a little disappointed when it turned up and was only the camera and waterproof case. A rather large whinge to CEX and turns out there was a storage case and a some accessories that hadn't been sent out so I'm much happier now that they sent them out. Accessories are pretty pricey so I'm wondering if accessories are standard with all these cameras mount wise?