Final answer:
The local police department in Alpha Township has primary investigative responsibility for crimes committed there, but the county sheriff's department may take over if the township lacks its own police. State or federal agencies can also be involved in specific circumstances.
Step-by-step explanation:
If a crime is committed in Alpha Township within the County of Beta, the primary investigative responsibility typically falls on the local law enforcement agency, which, in this case, would be the Alpha Township's police department. If the township does not have its own police, then the responsibility may fall to the county sheriff's department. For certain types of crimes, state or federal law enforcement agencies might also become involved, particularly if the crime involves violations of state or federal law, or if the local agencies request assistance.