No Inspector does the investigation. Ever. They manage other people doing the work, but they are not out taking statements, doing door to door or reviewing CCTV for example.
Constables (and the newish civilian investigators) do 99% of the actual work in the Police. Uniformed Sgts to be fair to them will try to be out working with their teams on patrol etc, but they don't carry workloads of investigations.
The odd DS may do interviews, but they'll usually be managing the interviews done by the constables. In some specialist units they will be leading the charge much more regularly but Inspectors are still not usually out with the team on jobs unless they need oversight on the ground.
As a DC I was certainly investigating murders, rapes, etc - the more senior you are, the more management heavy your role is. There are specific exceptions to this such as the Superintendents involved in sneaky beaky work.