This studio portrait has the word "Wanted" on the back, which suggests that it may have been arranged by the Royal Navy. As there are no badges on his uniform, it was probably taken during his first spell on HMS Victory from 27 Jan 1907 to 22 Jun 1907. The photo was coloured by Mum's boyfriend, Sidney Chapman, which I think is rather a shame as it makes him look as though he is taking part in an Amateur Dramatic production.
Frederick is wearing the hatband of HMS Topaze on which he served from 10 August 1909 to 31 July 1912. On his left arm is his Able Seaman's chevron and on his right arm his Gunner's badge. He is with his brother, Bob, who was a gunner in the Royal Artillery and was probably based at Woolwich, which is where this photo may well have been taken.