Hi,
an Londoner I know, who had a brother in the police many decades ago, is sure that bobby helmets were actually made especially/ or fitted especially for light motorcycle use.
There are many photos on Google images of police on bikes, scooters and motorcycles wearing the bobby helmets.
It was different in that it had a beat/foot chin strap, as well as a safety triple buckle chin strap for wear when on the cycle, as well as reinforced cork on the inside, much like a para or despatch riders helmet from the military. Can anyone provide more info ? dates, decades, departments issued these ? Thanks !