She did interfere, however, with the lives of Sheila,Jeremy, Colin and the twins and seemed to have free entrance to Moreshead Mansions whenever it suited. She must have had a guilty conscience,hence the heartfelt letter to be opened after her death which contained an apology of sorts, yet was too little too late to save the family from its fate.
Many mothers can be accused of 'interfering' by their children rightly or wrongly but in their eyes they are 'helping' and mean well, however annoying they may be.

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June was Sheila's mother so why would she not be allowed free access, especially when bearing food and money to pay the bills?
Yes, June may have known she was overbearing and judgemental but the fact she wrote such a letter shows her love for her children in spite of her mistakes and an awareness of her own shortcomings.
It is rather touching imo but who am I to judge, I wasn't there.