"To hold an inquiry" means to conduct or carry out an official investigation, typically done by a government or professional body, in order to gather information and find out more about a particular topic or event. Full definition
The committee is holding an inquiry into how well the government is pursuing its stated aim of achieving «gender equality and empowerment of all women and girls». (totalpolitics.com)
The assembly committee is holding an inquiry into the emotional and mental health of children and young people. (bbc.com)
The decision - though good as you credit - to hold the inquiry in public is allowing too much space for Cameron to manoeuvre, just like the public spending divide in the party has allowed space for George Osborne to appear on top of things, despite his expenses claims - which news of just disappeared into thin air - and his party's plans to cut spending on public services anyway. (nextleft.org)