That's not really correct. The first iteration of skinheads in Britain, late 60s, were a subculture with mod (white rockers) and rude boy (Jamaican immigrants) influences. Colloquially, reggae in the late 60s was referred to as skinhead reggae because skinheads were the biggest consumers of Jamaican music outside of the Caribbean. From an ideological standpoint, they were all over the place (benevolent toward blacks, with some 2nd gen Jamaicans in their ranks, but often hostile toward Pakistani immigration) but certainly not nazi or white power affiliated.

The National Front in the 70s started co-opting skinheads as they saw the disaffected youth as a means to rise in power. This is the birth of the nazi skinhead movement as we know it (aped later in America by shitheads like Tom Metzger in Fallbrook, CA), which coincided with racist bands gaining popularity like Skrewdriver or Combat 84.

However, most oi and street punk was antithetical to the National Front world view. In response, many prominent skinheads who were either apolitical to far Left (including Roddy Moreno of The Oppressed) started SHARP in response to the rise of skins identifying with National Front.

So that's the basic history.