English question, please help?

Change it to: "to investigate further".
This sounds much better ----- The police will investigate the issue further.

leave out the 's' at the end of details.

The statement is too wordy. "The police will investigate further." That's all that needs to be said; the sentence makes no sense taken out of context and would have to relate to an earlier statement that would identify the issue under investigation. Also, if the investigation is to be carried further then it would already be in progress.

look into it in more detail
or
look further into it, both are better

Nope!

Investigate sth for further details.

or "Further investigate sth" is ok too.