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Update 3: There is another interesting reference with relevance to the main theme of this post, namely, relations between FFTs for invariant theories of different groups which is how I understand the Erlangen Program. It is the recent preprint by Deligne, Lehrer and Zhang inspired by the classic article by Atiyah, Patodi and Singer where the FFT for the orthogonal group is deduced from of that of the general linear group.

If this were true then the FFT for $G=GL(V)$ would be the mother of all FFT's. The statement works for going from $GL(n)$ to $SL(n)$ and taking $A$ to be the epsilon Levi-civita tensor. For othogonal and symplectic groups one takes for $A$ a symmetric or antisymmetric form. But it works also if $G=SL(2)$ and $A=e_1=(1, 0)^T$ so that $H$ is the group of upper triangular matrices (I guess for $n>2$ one would need $A=e_1, e_1\wedge e_2,\ldots$). I thought the meta-theorem was mentioned in Franz Meyer's review, but in fact the only case he explicitly talks about is the last one. He calls that the study of "peninvariants" which is the same as seminvariants sources of covariants of binary forms etc.

If this were true then the FFT for $G=GL(V)$ would be the mother of all FFT's. The statement works for going from $GL(n)$ to $SL(n)$ and taking $A$ to be the epsilon Levi-civita tensor. For orthogonal and symplectic groups one takes for $A$ a symmetric or antisymmetric form. But it works also if $G=SL(2)$ and $A=e_1=(1, 0)^T$ so that $H$ is the group of upper triangular matrices (I guess for $n>2$ one would need $A=e_1, e_1\wedge e_2,\ldots$). I thought the meta-theorem was mentioned in Franz Meyer's review, but in fact the only case he explicitly talks about is the last one. He calls that the study of "peninvariants" which is the same as seminvariants sources of covariants of binary forms etc.

Update 2: A better "classical" reference (than Klein's of Meyer's) on the above method is the book by Turnbull "The Theory of Determinants, Matrices and Invariants". In Chapter XXI (in the 1960 Edition) he calls such FFT's "Adjunction Theorems", see in particular Sections 10 and 11.