Depends on what wave of skinheads you're talking about. Skinheads emerged in the UK in working class urban communities in the late sixties. They were influenced by the early sixties mod scene, the Jamaican rude boy scene, a strong identity towards their working class roots that alienated them against the government and the then middle class hippie lifestyle. As they were in working class areas in urban Britain, they were also rubbing shoulders with working class black people who brought with them their West Indian music and dress style. Ska Music and reggae were big influences on the tastes of first wave skinheads. In this way, it's ironic that the second wave of skinheads in the late seventies and beyond got involved with fascist politics, considering its working class multicultural roots.

I'm not sure you can say that skinheads were an offshoot of punk, at least not the first wave of skinheads, as skinheads predate punk by nearly a decade. As I mentioned above, the first skinheads were interested in ska, reggae and other music from West Indian roots. They were more an offshoot of the early sixties mods, with added interest in black working class styles and music.

The second wave came around when punk was in the ascendency in the late seventies, and that is where Oi music is based on. But Oi and the second wave's interest in fascism is certainly not what skinheads were originally about. The birth of Oi was convenient in a way for the likes of the fascist National Front in the UK, who were heavily recruiting amongst the skinheads in the late seventies, as it pulled skinheads away from that "problematic" (for the fascists) black music. I mean, there'd be a conflict of interest if you're heavily influenced by black music and styles and yet want to "send the foreigners back where they came from". In many ways Oi was a betrayal of the skinhead scene.