Repairing lateral house basement failures can be conducted inside or out.
In general, the wall failed due to lateral pressure. The repair must be stronger than the old wall. When the failure was caused by impact or vertical load, the old wall was too weak, even though it served until it failed.
The steps to repair are to unload the wall, straighten, add stiffeners, water proof, and backfill. The design must account for the lateral pressure, construction lateral pressures, surcharge loading, and hydrostatic. Unreinforced concrete wall, frequently used in residential cannot be numerically justified when backfill, hydrostatic, construction and surcharge are placed on them. In short, when we do the calculations, they fail often. In reality, they fail often.
It is a simple design of a beam column problem that the code ignores by using a defective prescriptive method which ignores surcharge, hydrostatic and construction loads. It also assumes back fill soils can be compacted without providing lateral pressure on the walls. In an effort to save money, the code is allowed to stand. The reason is the code is effectively controlled by home builders, who are aware of the cost and issue, and can gently lay in fill against the basement and allow settlement to occur. Repairs cannot allow the time for settlements. As a result, repairs must be considerably stronger that the original construction.