Isosceles Triangle Containers - an inside-out variant

We continue from this earlier post
Ref:
1. https://arxiv.org/abs/2001.09525
2. https://arxiv.org/abs/2205.11637

The above two papers address the questions of finding/characterizing the smallest enclosing and largest embedded isosceles triangles of a given general triangle (Note: 'smallest' and 'largest' may be defined with respect to either area and perimeter).

Let us record the inside-out versions of the same questions:
Given any isosceles triangle T to find that smallest triangle for which T is the smallest containing isosceles triangle. Likewise, find the largest triangle for which T is the largest contained(embedded) isosceles triangle.

The guess is that the perimeter versions of this question could be more difficult as opposed to the area versions.
Note: One can pose the same question with 'isosceles' replaced with 'right'. And of course, all these 'oriented container' investigations could have interesting variants in hyperbolic geometry.

One can generalize the question thus: Given an isosceles triangle T, to find the smallest(largest) convex region for which T is the smallest containing(largest embedded) isosceles triangle.

One could also ask for specific pairs (general triangle, isosceles triangle) that maximize the difference between container and containee.

A somewhat related overflow discussion
Updates to follow...