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I think it's just Cyclescheme administering it on behalf of your employer. I can't see how the asset (the bike) could be transfered to Cyclescheme in a way that they could continue to make money out of it otherwise.
[EDIT] Actually, I guess Cyclescheme just hire it from your employer in the same way that you do for the first year. At any point after that Cyclescheme have an option to buy it from your employer (if you want to) at the price determined by the HMRC tables, and are then free to sell it on for that same value to you. That keeps it all clean and easy for the employer. Cyclescheme have already made their money out of the cut of the original voucher price (the bike shop only receives 90% of the voucher value I believe).
[EDIT2] Seems to be backed up by http://www.cyclescheme.co.uk/employers/employer-updates/hmrc-update
"9. If the employment status of a scheme participant changes, what happens?
During the initial hire period, if Cyclescheme are notified of a change of employment status, we will contact the employee with their end of hire options at the point we are informed.
During the extended use period, the employer has no responsibility to contact Cyclescheme, as the agreement is between Cyclescheme and the employee. If the employee changes jobs the agreement is still valid."
So you just need to stay in your job for the first 12 months. You can then run the Extended Use Agreement with Cyclescheme and not worry about changing jobs.
This is assuming that your employer uses Cyclescheme to do this. There are other schemes (and direct employer schemes too).
Is this the case? So far as I can see, my extended use period (years 2-4) is with Cyclescheme, not my company. This is the T&Cs from my Agreement. Section 8 seems to cover it.
This may only relate to Cyclescheme and is from a few years ago so it could have changed.