This is an interesting dilemma, but one which can be resolved fairly easily, in sha’ Allah. Most children are affected by whatever the current, popular pastime is during their childhood. In this case, it seems that your children really like to play computer games. But looking at the ages of your children, it seems odd that the younger children would go anywhere by themselves if the oldest one is supposed to be looking after them. Is it the oldest one that wants to go out and ends up taking the other, younger, children out to play computer games? If so, then you need to focus your attention on the eldest child as the guardian or caretaker in your absence. If only one of them is sneaking out of the house to play computer games, then you can focus your attention on that specific child.
Suppose the eldest child is the one that is going out, we suggest that you have a private conversation with him. The gender of the child is unclear from your message, but for the sake of the response we will assume it is a male. We are writing in general, so that the response will apply even if the eldest is a girl.
You should not be too harsh on the oldest child, but you should be firm. For example, share your feelings about trust with him. Let him know that he is entrusted with the welfare of the family in your absence. Not in a comprehensive sense, but only in the sense that he has to make sure they have a snack after school, they do their homework, they relax for a while, and, most importantly, they are safe. By letting the children sneak out to play computer games against your wishes, or taking them himself, your eldest child is violating your trust. He needs to know how that makes you feel. If the children have been lying for sometime, you should address that with all of them, so that they realize that you know the truth now and that there will be consequences if they lie in the future. The last thing you want, is for the children to begin to cover for each other’s mistakes by lying to you.
Finally, if you, as a parent, find that the computer games the children are playing are not violent and are acceptable in terms of the message they convey, then by all means, arrange a schedule so that the children know when you will take them. This does not have to be every week, but at least the children should understand that you will take them and that they do not have to sneak out to go and play.
We hope that this helps. The bigger issue here might be for you to ask an older relative or close friend to baby-sit while you and your husband are at work. Of course, it will cost you some money to get a babysitter, but at least then you can hold an adult responsible for looking after your children. The 13-year-old might have his hands full with the other kids. Don’t forget that he is a teenager who has to address his own growing pains. Please review the situation with your husband and consider hiring an adult babysitter.