The Short Answer
A father with parental responsibility can remove a child from the mother's care for short periods -- for example, for a contact visit. But he cannot remove the child permanently without the mother's consent or a court order. Doing so without consent or a court order can amount to child abduction.
Parental Responsibility and Day-to-Day Decisions
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What Do I Do Next? -- FreeWhere both parents have parental responsibility, day-to-day decisions about the child's care can be made by whoever the child is with. But significant decisions -- where the child lives, schooling, medical treatment, taking the child abroad -- require the agreement of all those with parental responsibility or a court order.
What the Law Says About Removal
The Child Abduction Act 1984 makes it a criminal offence for a parent to take a child under 16 out of the UK without the other parent's consent or a court order, or to take a child in breach of a court order. Removing a child from the other parent's care in breach of a court order is also contempt of court.
If the Father Takes the Child Without Consent
If a father takes a child without the mother's consent in a way that goes beyond normal contact arrangements, the mother can apply urgently to the family court for a child arrangements order and a prohibited steps order. The court can act within hours in genuine emergencies.
The Correct Route
If a father is unhappy with the current arrangements and wants the child to live with him, the correct route is to apply to the family court for a child arrangements order. Taking matters into his own hands is likely to damage his position in any subsequent court proceedings.