A relaxed evaluation subgroup

Let $f:X\to Y$ be a pointed map between connected CW-complexes. As a generalization of the evaluation subgroup $G_*(Y,X;f)$, we will define the {\it relaxed evaluation subgroup} ${\mathcal G}_*(Y,X;f)$ in the homotopy group $\pi_*(Y)$ of $Y$, which is identified with ${\rm Im} \pi_*(\tilde{ev})$ for the evaluation map $\tilde{ev} :map(X,Y;f)\times X\to Y$ given by $\tilde{ev} (h,x)=h(x)$. Especially we see by using Sullivan model in rational homotopy theory for the rationalized map $f_{\Q}$ that ${\mathcal G}_*(Y_{\Q},X_{\Q};f_{\Q})=\pi_*(Y)\otimes \Q$ if the map $f$ induces an injection of rational homotopy groups. Also we compare it with more relaxed subgroups by several rationalized examples.