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We prove that whenever the selfmapping <inline-formula><math display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>M</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>M</mi><mi>p</mi></msub><mo>)</mo></mrow><mo lspace="0pt">:</mo><msup><mi>I</mi><mi>p</mi></msup><mo>→</mo><msup><mi>I</mi><mi>p</mi></msup></mrow></semantics></math></inline-formula>, (<inline-formula><math display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>M</mi><mi>i</mi></msub></semantics></math></inline-formula>-s are <i>p</i>-variable means on the interval <i>I</i>) is invariant with respect to some continuous and strictly monotone mean <inline-formula><math display="inline"><semantics><mrow><mi>K</mi><mo lspace="0pt">:</mo><msup><mi>I</mi><mi>p</mi></msup><mo>→</mo><mi>I</mi></mrow></semantics></math></inline-formula> then for every nonempty subset <inline-formula><math display="inline"><semantics><mrow><mi>S</mi><mo>⊆</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>p</mi><mo>}</mo></mrow></semantics></math></inline-formula> there exists a uniquely determined mean <inline-formula><math display="inline"><semantics><mrow><msub><mi>K</mi><mi>S</mi></msub><mo lspace="0pt">:</mo><msup><mi>I</mi><mi>p</mi></msup><mo>→</mo><mi>I</mi></mrow></semantics></math></inline-formula> such that the mean-type mapping <inline-formula><math display="inline"><semantics><mrow><mrow><mo>(</mo><msub><mi>N</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>N</mi><mi>p</mi></msub><mo>)</mo></mrow><mo lspace="0pt">:</mo><msup><mi>I</mi><mi>p</mi></msup><mo>→</mo><msup><mi>I</mi><mi>p</mi></msup></mrow></semantics></math></inline-formula> is <i>K</i>-invariant, where <inline-formula><math display="inline"><semantics><mrow><msub><mi>N</mi><mi>i</mi></msub><mo>:</mo><mo>=</mo><msub><mi>K</mi><mi>S</mi></msub></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>S</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>N</mi><mi>i</mi></msub><mo>:</mo><mo>=</mo><msub><mi>M</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula> otherwise. Moreover <inline-formula><math display="inline"><semantics><mrow><mo movablelimits="true" form="prefix">min</mo><mrow><mo>(</mo><msub><mi>M</mi><mi>i</mi></msub><mo lspace="0pt">:</mo><mi>i</mi><mo>∈</mo><mi>S</mi><mo>)</mo></mrow><mo>≤</mo><msub><mi>K</mi><mi>S</mi></msub><mo>≤</mo><mo movablelimits="true" form="prefix">max</mo><mrow><mo>(</mo><msub><mi>M</mi><mi>i</mi></msub><mo lspace="0pt">:</mo><mi>i</mi><mo>∈</mo><mi>S</mi><mo>)</mo></mrow><mo>.</mo></mrow></semantics></math></inline-formula> Later we use this result to: (1) construct a broad family of <i>K</i>-invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.