Using this phrase often implies a significant amount of trust in the person being given freedom. The manager decided to give his team their head on the project.
Pro Tip 2/3
Responsibility Aspect
It suggests the person is now responsible for their choices and actions. With the new responsibilities, she was given her head to lead the team.
Pro Tip 3/3
Potential Risks
Be aware that it can indicate risks since allowing autonomy can lead to unexpected outcomes. The director gave the actors their head, risking an unpredictable performance.