It is well-known that any left exact functor $F : \mathbf{A} \to \mathbf{B}$ between abelian categories yields a sequence of derived functors $F^n : \mathbf{A} \to \mathbf{B}$ such that for any short exact sequence $0 \to A \to B \to C \to 0$ ín $\mathbf{A}$ we get a long exact sequence $0 \to F(A) \to F(B) \to F(C) \to F^1(A) \to F^1(B) \to F^1(C) \to F^2(A) \to \dots$.
This applies in particular to the inverse limit functor $\varprojlim : \mathbf{Ab}^I \to \mathbf{Ab}$, where $\mathbf{Ab}$ is the category of abelian groups and $\mathbf{Ab}^I$ is the category of inverse systems of abelian groups indexed by a directed set $I$ (or more generally by a cofiltered small category $I$).
In
Bousfield, A. K., & Kan, D. M. (1972). Homotopy limits, completions and localizations (Vol. 304). Springer Science & Business Media
which dates from 1972 one can find a definition of the first derived limit functor $\varprojlim^1 : \mathbf{Gr}^I \to \mathbf{Set_0}$, where $\mathbf{Gr}$ is the category of groups and $\mathbf{Set_0}$ is the category of pointed sets. This is certainly a sort of adhoc-construction which does not generalize to produce higher derived functors. As far as I know, this is the first occurence of this functor in the literature.
My question is: Are there earlier papers introducing $\varprojlim^1 : \mathbf{Gr}^I \to \mathbf{Set_0}$, perhaps in the special case $I = \mathbb{N}$?
There are some papers which seem to be relevant, e.g.
Gray, B. I. (1966). Spaces of the same n-type, for all n. Topology, 5(3), 241-243
but a closer look shows that they only give a definition for inverse systems of abelian groups.