Does this make sense, is this test appropriate in order to see how responses differ among more then 2 different groups? I have no experience working with SPSS. Is there any chance someone can help me to find appropriate tests for what I need (I can explain better) to do and to interpret the results?
Krustal-Wallis is appropriate for this use-case if we "simply" want to identify if at least one of the groups in our overall sample is different from at least one other group. i.e. we will get the $p$-value for the null hypothesis that the data in each group comes from the same distribution. In that sense, KW is fine first step.
Now, as we potentially care to identify which group comes from a different distribution we need to have a principled way to perform multiple group $A$ vs group $B$ comparisons. They are many different procedures to perform these multiple comparisons in theoretically coherent way, e.g. Scheffe's procedure, Bonferroni correction, Tukey's Honestly Significant Difference procedure, Dunn-Šidák's cor, Holm-Bonferroni correction, etc. Ultimately all procedures try to control for what is called Family Wise Error Rate (the probability of making at least one type I error (False positive) in our family of tests - see this (in)-famous XKCD "green jelly beans" cartoon for a nice cartoonish description). As a starting point, I would suggest you try Holm-Bonferroni (sometimes called simply Holm or Holm step-down) correction. It is available in most statistical packages and it can be easily explained as directly controlling FWER while being more powerful than the "ordinary" Bonferroni procedure.