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A kite is a triangle with a tail consisting of a single edge. A kite system of order \(n\) is a pair \((X,K)\), where \(K\) is a collection of edge disjoint kites which partitions the edge set of \(K_n\) (= the complete undirected graph on \(n\) vertices) with vertex set \(X\). Let \((X,B)\) be a block design with block size 4. If we remove a path of length 2 from each block in \(B\), we obtain a partial kite-system. If the deleted edges can be assembled into kites the result is a kite system, called a \emph{metamorphosis} of the block design \((X,B)\). There is an obvious extension of this definition to \(\lambda\)-fold block designs with block size 4. In this paper we give a complete solution of the following problem: Determine all pairs \((\lambda, n)\) such that there exists a \(\lambda\)-fold block design of order \(n\) with block size 4 having a metamorphosis into a \(\lambda\)-fold kite system.